Constant Power Model in Arm Rotation—A New Approach to Hill’s Equation

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Constant Power Model in Arm Rotation—A New Approach to Hill’s Equation Rahikainen, Ahti; Virmavirta, Mikko Rahikainen, A., & Virmavirta, M. (2014). Constant Power Model in Arm Rotation—A New Approach to Hill’s Equation. World Journal of Mechanics, 4(6), 157-169. https://doi.org/10.4236/wjm.2014.46018 2014 World Journal of Mechanics, 2014, 4, 157-169 Published Online June 2014 in SciRes. http://www.scirp.org/journal/wjm http://dx.doi.org/10.4236/wjm.2014.46018 How to cite this paper: Rahikainen, A. and Virmavirta, M. (2014) Constant Power Model in Arm Rotation—A New Approach to Hill’s Equation. World Journal of Mechanics, 4, 157-169. http://dx.doi.org/10.4236/wjm.2014.46018 Constant Power Model in Arm Rotation—A New Approach to Hill’s Equation Ahti Rahikainen, Mikko Virmavirta Neuromuscular Research Center, Department of Biology of Physical Activity, University of Jyväskylä, Jyväskylä, Finland Email: [email protected] Received 19 March 2014; revised 18 April 2014; accepted 16 May 2014 Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract The purpose of this study was to further develop the constant power model of a previous study and to provide the final solution of Hill’s force-velocity equation. Forearm and whole arm rotations of three different subjects were performed downwards (elbow and shoulder extension) and upwards (elbow and shoulder flexion) with maximum velocity. These arm rotations were recorded with a special camera system and the theoretically derived model of constant maximum power was fitted to the experimentally measured data. The moment of inertia of the arm sectors was calculated using immersion technique for determining accurate values of friction coefficients of elbow and whole arm rotations. The experiments of the present study verified the conclusions of a previous study in which theoretically derived equation with constant maximum power was in agreement with experimentally measured results. The results of the present study were compared with the mechanics of Hill’s model and a further development of Hill’s force-velocity relationship was derived: Hill’s model was transformed into a constant maximum power model consisting of three different components of power. It was concluded that there are three different states of motion: 1) the state of low speed, maximal acceleration without external load which applies to the hypothesis of constant moment; 2) the state of high speed, maximal power without external load which applies to the hypothesis of constant power and 3) the state of maximal power with external load which applies to Hill’s equation. This is a new approach to Hill’s equation. Keywords Arm Movement, Force-Velocity Relationship, Muscle Power, Hill’s Equation 1. Introduction Hill’s force-velocity relationship of skeletal muscle (Figure 1) [1] [2] is one of the most essential equations of A. Rahikainen, M. Virmavirta 158 Figure 1. Hill’s force-velocity relationship with corresponding power-velocity curve (dashed gray curve), where F0 is maximum isometric force or force with zero velocity, v0 is the highest possible velocity, a and b are constant force and constant velocity. Maximum power P0 is typically found at about 30 % of v0 [4]. In rotational movement torque M corresponds to force F and angular velocity corresponds to velocity v in Hill’s equation. muscle mechanics and it has been an object of biomechanical studies for years (e.g., [3]-[6]). In muscle mechanics, this relationship is often presented by Hill’s equation ( )( ) ( ) 0 F a v b bF a+ += + , where F is current muscle force at current shortening velocity of contraction, a is constant force and b is constant velocity, F0 is the maximum isometric muscle force, i.e., the maximum force that muscle can develop at a given constant length, and v is velocity [1] [2]. For comparison between different muscles b is best expressed in terms of b/l0, where l0 is the standard length of muscle. This equation was based on laboratory measurements with a Levin-Wyman ergometer in which the activated muscle was released at a suitable speed in an isolated muscle condition. The obtained constant velocity was then plotted against the observed tension. Force measured from skeletal muscle during maximum tension depends on several internal and external factors which have been thoroughly reviewed (e.g., [3] [7] [8]). Matsumoto [9] mentioned that because almost all the isotonic data have been restricted to one muscle length l0, the maximum length with almost no resting tension, and the velocities measured are those initial values when the load begins to move. Examining the length region, l ≤ l0, for an isotonically contracting muscle, Matsumoto [9] found that the constants a/F0 and b/l0 remained fixed throughout the range of lengths over which the shortening takes place. In contrast to Hill’s isolated muscle preparations, force (F) of the involved muscles in rotational movement creates a moment (M = r × F) about the joint. The length of the muscle’s moment arm (r) changes as the rotational movement proceeds about the joint axis. This rotation movement is the combined effect of the forces of several different muscles. However, it is difficult to determine the contribution of each muscle on force production due to many different factors, and also to determine the torque about the joint. Several extensor and flexor muscles were used by Raikova [10] in her model of the flexion-extension motion in the elbow joint. Furthermore, the force of a skeletal muscle is an accumulation of forces generated by active motor units belonging to this muscle [11]. Raikova et al. [12] mentioned that access to each motor unit (MU) is impossible, and the recruitment and force developing properties of all individual MUs cannot be known. In this paper [12] the process of learning fast elbow flexion in the horizontal plane was simulated and the result was compared with experimentally measured data. Previous experiments of Rahikainen et al. [13] were based on the theory of arm rotations including four phases: 1) start of the motion; 2) phase of constant maximum rotational moment; 3) phase of constant maximum muscular power; 4) stopping of the motion. It was assumed that the muscular system is able to transfer only a certain amount of chemical energy during the time of contraction and therefore, arm rotation must have maximum power that cannot be exceeded. It was also assumed that the maximum power acts within a certain range of velocity which was considered as constant maximum power and this is possible only when the velocity is F/P v F 0 v 0 a b P 0 A. Rahikainen, M. Virmavirta 159 high enough. The velocity of the motion increases to the point where the maximum power occurs and acting rotational moment is less than the maximum moment. Consequently, power remains constant as the angular velocity increases and the moment decreases. The present study continues the experiments of Rahikainen et al. [13] and further develops its theory of mechanics. The previous study presented a constant power model and its validity and accuracy of results were assessed. In the present study new arm rotation experiments of three subjects were captured with two different recording systems. More accurate calculations of moments of inertia and a new more effective determination of the matched range of measured and theoretical angular velocity curves were used. Also a new approach to Hill’s equation was presented. In the left side of Hill’s equation ( )( ) ( ) 0 F a v b bF a+ += + the constant a has the dimensions of force and b the dimensions of velocity, otherwise addition is impossible. Therefore, (F + a) is force and (v + b) is velocity, and force multiplied by velocity is power as can be seen in Figure 1. The term (F + a) (v + b) is muscles’ total power including Fv, which is the power of moving the external load. The right side of the equation, b (F0 + a), includes only constants in this regard the equation can be considered as a constant power model. However, the constant power of Hill’s equation presented in this paper is not the above mentioned power of Hill’s original curve as it is usually considered in biomechanics, but it is the sum of three different power components (see Discussion and Conclusions). The constant power model of this study acts during high speed movements with no external load, where Hill’s equation does not seem to fit the experimental points ([2] p. 32, Figure 3) very well. As an explanation for this mismatch Hill mentioned that “sharp rise at the end of the curve in the region of very low tension was due to the presence of a limited number of fibers of high intrinsic speed and no such equation could fit the observed points below P/P0 = 0.05”. The present model is based on the muscular system’s ability to transfer chemical energy and, therefore, it is not necessary to know the contribution of the individual muscles involved. The purpose of the present study was to examine how the theoretical constant power model which was first used in linear motion [14] and later applied to rotational motion [13] fits the measured angular velocity curves of arm rotation experiments. The further purpose of the study was to determine how Hill’s equation functions as a constant power model and to compare Hill’s equation with the model of the present study. 2. Materials and Methods In the present study, the measurements of arm rotations (Subjects S1 and S2) were recorded by a special camera system [15] [16]. Figure 2 shows the principle of the system where angular velocity was calculated with the Formula: SRT ϕ =∆∆  (1) where R is arm length, ΔS is the distance between two successive measuring points and ∆T is time increment. Additional elbow flexion and extensions were performed for one Subject S3 by using Vicon motion analysis system with 8 cameras. This system made it possible to use higher frame rates (300 Hz). Subjects were normal healthy people representing different age and physical activity background (S1: 62 years, 1.80 m, 82 kg; S2: 35 years, 1.69 m, 73 kg, aikido training, weight lifting and S3: 25 years, 1.83 m, 70 kg, high jumping). The accuracy of the special camera system has been described in references [13] [15]-[17]. The angular velocity-time curves were drawn (with a line thickness of 0.5 mm) on the squared paper, where 1 mm corresponded to an angular velocity of 0.1 rad/s and time of 1 ms giving the accuracy of velocity curves. The theoretical and measured angular velocity curves coincided within the distances of 35 - 70 mm, which was enough for the verification of the constant power model in the arm rotation experiments. Slight oscillations at the beginning of the movement did not exist within the constant power phase and, therefore, the verification of the constant power model was possible in all experiments. The accuracy of the Vicon recording device, verified by calculating the root-mean-square error for each camera, ranged from 0.06 to 0.17 mm during calibration. 2.1. Model of Arm Rotation The model used in the present study is constructed according to Newton’s II law and it was first used in linear motion [14] and then applied to rotational motion [13] by Rahikainen et al. The theory of arm rotation is as follows: At the beginning of the movement, angular velocity is naturally zero and it takes some time to generate force. At this early phase of motion, elastic properties of muscle-tendon complex has influence on the motion, but at the state of full tension, these elastic elements have no further dynamic effect. Thereafter it can be A. Rahikainen, M. Virmavirta 160 Figure 2. Example of elbow flexion. Rotation angles between two arm images (0 - 200 ms) are given in 20 ms increments according to light marks. It is noteworthy that the moment arm of the muscles involved in elbow flexion changes during the movement and, therefore, the muscles’ contribution to power changes as well. assumed that a maximum muscle force takes action and within rotational motion maximum rotational moment acts as well. At this phase of motion there is a constant value of glide friction acting. Because the muscle system is able to transfer only a certain quantity of chemical energy during the time of contraction, there must be a constant maximum power, which the muscle is able to generate within a certain range of velocity. As the velocity increases the motion reaches the point where the maximum power takes action and acting rotational moment is less than the maximum moment. This way power remains constant as the angular velocity increases and moment decreases. Now, liquid friction, directly proportional to velocity, is acting. The constant glide friction decreases as forces at the joint decrease and it becomes negligible. During the constant power phase of the model rotational moment is moment of inertia multiplied by angular acceleration which equals the moment generated by muscle force minus the moment generated by inner friction of muscle. The effect of gravitational force is added afterwards (see Paragraph 2.3). The model of arm rotation is the equation of motion: d d P IC T ϕϕ ϕ = −   (2) where Moment of inertia in arm rotation I Angular velocity ϕ  Power generated by arm muscles P Time T Moment generated by muscle force P ϕ  Moment generated by inner friction of muscle C ϕ  Coefficient of friction C Inner friction of muscle is liquid friction inside muscle, which is directly proportional to velocity. The same liquid friction was also used in the study of Rahikainen et al. [17] which was initially adopted from Alonso and Finn [18]. It was assumed that, initially, movement proceeds at a constant maximum moment and then moment generated by the muscle force ( P ϕ  in Equation 2) is constant. It was also assumed that, as velocity increases, 200 180 160 140 120 100 80 60 40 20 0 A. Rahikainen, M. Virmavirta 161 movement proceeds at constant maximum power at a certain range of velocity and then the power P in Equation 2 is constant. In order to determine the validity of this hypothesis, Equation 2 was solved for angular velocity-time function and this equation was employed for validity determination. The solution of Equation 2 from the previous study [13]: 2 1e CT I P C ϕ −  = −     (3) 2.2. Calculation of Moment of Inertia The mass distribution of the subject’s arm sectors differed from the average values of mass tables in the literature. Therefore, the mass distribution of the arm sectors were determined by sinking the arm sectors into water. The masses of the arm sectors were calculated by multiplying the over flowed water volume with the corresponding arm sector density. The length of the subject’s whole arm was measured with fist clenched, and the lengths of arm sectors (hand, forearm and upper arm) were measured. According to Winter [6] the arm sector densities (kg/l) were as follows: hand 1.16, forearm 1.13, upper arm 1.07. Definition of moment of inertia: 2 dI rm=∫ (4) where dm is rotating mass and r is the distance of rotating mass from the rotational axis. Derivation of the moment of inertia about the end of arm sector assuming even mass distribution: 2 22 2 1 00 0 d d dd d3 LL L m m mL I rm r r rr rL = = = = ∫∫ ∫ (5) where m is the mass of rotating arm, and L is the length of rotating arm. Derivation of the moment of inertia about the center of the arm sector assuming even mass distribution: 233 2 2 233 2 d3 12 22 L L m L L mL I rm L −  = = +=   ∫ (6) Because the distribution of mass in the arm sector is not even, the moment of inertia of an additional mass was calculated with Formula: 2 3 I mr= (7) where m is additional mass of the arm sector and r is the estimated distance of the center of gravity of additional mass from the rotation axis. Subjects’ arm sector distances, lengths and masses are presented in Table 1. These values are then substituted in the above mentioned Equations 4-7 to calculate the final moments of inertia of forearm and whole arm rotations about the elbow and shoulder joint (example for S1 summarized in Table 2). Moment of inertia of forearm rotation Forearm Moment of inertia of major forearm mass about the elbow joint (m = 1.0 kg, L = 0.28 m): 22 2 11 1.0 0.28 0.0261kg m 33 mL I× = = = ⋅ Moment of inertia of additional forearm mass about the elbow joint (m = 0.24 kg, r = 0.07 m): 22 2 12 0.24 0.07 0.0012 kg mI mr==×= ⋅ Hand Moment of inertia of the hand sector about the center of mass (m = 0.58 kg, L = 0.11 m): 22 2 13 0.58 0.11 0.0006 kg m 12 12 mL I× = = = ⋅ Moment of inertia of the hand sector about the elbow joint (m = 0.58 kg, r = 0.34 m): 22 2 14 0.58 0.34 0.0670 kg mI mr==×= ⋅ A. Rahikainen, M. Virmavirta 162 Table 1. The estimated mass distribution of arm sectors. Forearm’s additional mass is due to the muscle’s mass distribution to the other end of arm sector. Distance from elbow joint (m) Distance from shoulder joint (m) Length (m) Mass (kg) Subject S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 Hand 0.11 0.11 0.12 0.58 0.52 0.58 center of mass 0.34 0.32 0.34 0.64 0.60 Forearm 0.28 0.26 0.27 1.00 0.90 1.00 center of mass 0.44 0.41 additional mass 0.07 0.06 0.07 0.37 0.34 0.24 0.22 0.24 Upper arm 0.30 0.28 0.33 2.14 2.53 Battery 0.37 0.35 0.67 0.63 0.26 0.29 0.29 Table 2. Summarized information for calculation of moment of inertia of forearm rotation (upper part) and whole arm rotation (lower part) for one subject. Moment of inertia is calculated about the elbow joint, center of mass (com) and shoulder joint. Segment m (kg) L (m) r (m) About Moment of inertia I (kg⋅m2) Forearm 1.00 0.28 elbow 2 11 3 mL I= 0.0261 Forearm (additional mass) 0.24 0.07 elbow 2 12 I mr= 0.0012 Hand 0.58 0.11 com 2 13 12 mL I= 0.0006 Hand 0.58 0.34 elbow 2 14 I mr= 0.0670 Battery 0.26 0.37 elbow 2 1b I mr= 0.0356 Total 0.131 Segment m (kg) L (m) r (m) about moment of inertia I (kg⋅m2) Upper arm 2.14 0.30 shoulder 2 21 3 mL I= 0.0642 Forearm 1.00 0.28 com 2 22 12 mL I= 0.0065 Forearm 1.00 0.44 shoulder 2 23 I mr= 0.1936 Forearm (additional mass) 0.24 0.37 shoulder 2 24 I mr= 0.0329 Hand 0.58 0.11 com 2 25 12 mL I= 0.0006 Hand 0.58 0.64 shoulder 2 26 I mr= 0.2376 Battery 0.26 0.67 shoulder 2 2b I mr= 0.1167 Total 0.652 Light marker battery Moment of inertia of the battery about the elbow joint (m = 0.26 kg, r = 0.37 m): 22 10.26 0.37 0.0356 kg m b I=×= ⋅ Total moment of inertia of forearm rotation about elbow joint 2 0.0261 0.0012 0.0006 0.0670 0.0356 0.131kg m++++= ⋅ A. Rahikainen, M. Virmavirta 163 Moment of inertia of whole arm rotation Upper arm Moment of inertia about the shoulder joint (m = 2.14 kg, L = 0.30 m): 22 2 21 2.14 0.30 0.0642 kg m 33 mL I× = = = ⋅ Forearm Moment of inertia of the major forearm mass about the center of mass (m = 1.0 kg, L = 0.28 m): 22 2 22 1.0 0.28 0.0065 kg m 12 12 mL I× = = = ⋅ Moment of inertia about the shoulder joint (m = 1.0 kg, r = 0.44 m): 22 2 23 1.0 0.44 0.1936 kg mI mr==×= ⋅ Moment of inertia of additional mass about the shoulder joint (m = 0.24 kg, r = 0.37 m): 22 2 24 0.24 0.37 0.0329 kg mI mr==⋅= ⋅ Hand Moment of inertia of the hand sector about the center of mass (m = 0.58, L = 0.11 m): 22 2 25 0.58 0.11 0.0006 kg m 12 12 mL I× = = = ⋅ Moment of inertia of the hand sector about the shoulder joint (m = 0.58 kg, r = 0.64 m): 22 2 26 0.58 0.64 0.2376 kg mI mr==×= ⋅ Light marker battery Moment of inertia of the battery about the shoulder joint (m = 0.26 kg, r = 0.67 m): 22 20.26 0.67 0.1167 kg m b I=×= ⋅ Total moment of inertia at whole arm rotation about shoulder joint 2 0.0642 0.0065 0.1936 0.0329 0.0006 0.2376 0.1167 0.652 kg m++++++= ⋅ According to the above mentioned calculations, the corresponding moments of inertia of forearm rotation and whole arm rotation for S2 were 0.110 and 0.551 kg⋅m2, respectively. As the total length of forearm and hand was the same for Subjects S1 and S3, it was assumed that the moments of inertia of these segments were also the same and, therefore, the moment of inertia of forearm rotation about the elbow joint for S3 was 0.131 kg⋅m2. The measurements with Vicon camera system for S3 were done with reflective markers and the corresponding moment of inertia without the battery was 0.095 kg⋅m2. 2.3. Effect of Gravitational Force on the Movement In forearm rotation, the effect of gravitational force is minor compared with maximum muscle forces and the moment induced by gravity r mgΣ× was omitted from the motion model (Equation 2). In whole arm rotation, this moment is added to the motion mechanics in the following manner: The power generated by this gravitational moment is ( ) r mg ϕ Σ×  , where mg is gravitational force of arm segments, r is distance of the center of gravity of segments from the rotation axis and ϕ  angular velocity of arm rotation. The theoretical angular velocity (Equation 3) and the measured angular velocity match within a very narrow velocity range and the power induced by gravity can be calculated as a constant factor. It is included in the power P according to the previous study [13]. 2.4. Determining the Matched Range of Measured and Theoretical Angular Velocity Curves Figure 3 shows the technique that was used to determine the matched range of measured and theoretical angular A. Rahikainen, M. Virmavirta 164 Figure 3. Example of the technique to find the matched range (A - B) of measured and theoretical angular velocity. The zero point of time for the theoretical angular velocity is at the intersection of the time-axis and the broken-line curve (see text for more information). The theoretical angular velocity curve (broken line) coincides with the measured curve (solid line) between A and B. velocity curves. Friction coefficient values (C) and power values (P) were obtained by fitting the theoretical angular velocity curves to the measured ones. These two curves coincide only if certain C and P values are used in the fitting process. The measured angular velocity values are shown as points on the velocity curve and the theoretical angular velocity (Equation 3) is shown as a broken line. The matched range was found by using the following iteration procedure: based on previous experiments, the randomized initial values (see Figure 3) within the matched range were selected for angular velocity at point 1 ( ) 16.8 rad s ϕ =  . A zero point on the time axis for the theoretical angular velocity was selected 0.050 s before point 1, which corresponds to 0.060 s on the time axis of the figure. Thereafter, the direction of iteration process was observed and after some iteration, the final theoretical angular velocity was drawn according to Equation 3 to match the measured velocity curve. The correct theoretical angular velocity was obtained with the zero point at 0.073 s on the time axis. The ratio of power and friction coefficient in Figure 3 is 2 2 1e CT I P C ϕ − = −  (8) The final theoretical angular velocity was drawn with moment of inertia I = 0.135 kg⋅m2, friction coefficient value C = 2.92 kg⋅m2/s and ratio of power and friction coefficient P/C = 354 1/s2. 3. Results Arm rotation experiments recorded by the camera system of Rahikainen [16] are presented in Figures 4-7 and experiments with Vicon motion analysis system in Figure 8. The theoretical angular velocity curves in Figures 4-8 are marked with broken lines and they coincide with the measured angular velocity curves (solid lines) between the points A - B, where movement proceeds at constant power. Initially, movement proceeds at constant acceleration, then liquid friction becomes influential and acceleration decreases just before the section A - B (constant power), which is finally followed by stopping of the movement. In general, the sections A - B are long enough to verify the existence of the constant power model. The measurements in Figure 8 with Subject S3, made by the Vicon motion analysis system, did not have a clear section of constant acceleration at the beginning of the movement. High oscillation in that section turned it indefinite. In the elbow extension, the oscillation is weak and the usual constant acceleration section can be distinguished at the movement initiation. It also seems that the constant acceleration section in Figure 6 has a similar oscillation as in Figure 8. The measured data in Figures 4-8 have been smoothed by the 6th order polynomial curve fitting. The used friction coefficient values varied between 2.8 - 3.1 kg⋅m2/s in forearm rotations and between 3.6 - 3.8 kg⋅m2/s in 50 200150100 T(ms) (rad/s) I0.135 kgm2 C2.84 kgm2/s P/C 321 1/s2 I0.135 kgm2 C2.92 kgm2/s P/C 354 1/s2 16.8 rad/s 1 A B