Dynamic balance of circles with one support point on pommel horse exercise

Abstract

Double-leg circles on pommel horse exercises require a high degree of dynamic balance. However, theoretical conditions for maintaining dynamic balance are unclear. The purpose of this paper is to propose a simple theoretical model of the dynamic balance of the circles, and to illustrate its qualitative properties. To this end, the body of a gymnast is simply modeled as one rigid body with one support point, and symmetric and constant-velocity circles are assumed in most analyses. The condition that the torques of wrist and shoulder joints are zero is assumed as a dynamic balance condition with minimum strength. A control law is proposed to demonstrate the motion. Various properties of the dynamic balance condition are analyzed as follows. In the symmetric circles, (1) as the period of a circle decrease, the heights and the radius of the toes increase, and (2) the rotation of the body around its longitudinal axis in the double-leg circle has the effect of lifting the toes. (3) The shoulder and wrist torques can change the pose around the dynamic balance condition. In asymmetric circles, (4) the radius of the center of mass (CoM) increases as the angular velocity of the CoM around the support point decreases, and (5) the body angle with respect to the horizontal plane increases as the upward acceleration of the CoM increases. Moreover, mechanical principles of the circles are discussed as follows. (6) The CoM motion during the symmetric circles of the simple model can be related to the conical pendulum. In the simple model, (7) the pommel reaction force should be parallel to the arm segment when the wrist torque and the shoulder torque are equal or zero, and (8) to change the vertical component of the angular momentum of the body around the support point, the wrist torque around the vertical axis is needed. These results provide theoretical and qualitative insights into understanding and improving pommel horse exercise.

Introduction

The double-leg circle is the most basic technique on the pommel horse exercises of the artistic gymnastics. However, it is also a complex motion, because it requires rotating the center of mass (CoM) of the body around a support point, and simultaneously rotating the body around the vertical axis and the longitudinal axis. Many studies have investigated details of the double-leg circles. Markolf et al.¹ measured force exerted on the pommel. Baudry et al.^2,3 and Grassi et al.⁴ performed motion measurements. Fujihara et al. performed motion and force measurements of the circles⁵ and the circles with a suspended aid⁶, and motion measurements of the cross circles⁷. Nawa et al.^8,9 performed motion, force, and electromyography measurements of the circles, and motion measurements of Flop technique¹⁰. Qian et al.¹¹ performed motion and electromyography measurements, and simulation by a multi-body model. Murai et al.¹² and Yamada et al.¹³ performed motion, force, and electromyography measurements, and estimation of muscle activities by a multi-body model. Murai et al. pointed out from the analysis of the principal moments of inertia that the gymnasts performed circles almost as if their entire body was a rigid body¹². However, there has been little discussion of the fundamental dynamics of the circles, while the details of the motion have been widely studied. The dynamics of the pommel horse exercise is very different from that of other exercises, and is still quite difficult to understand, even in terms of qualitative properties.

Also, many studies have focused on evaluating the quality of the double-leg circles. Grassi et al.⁴ evaluated diameters and deviations from circularity and from the horizontal plane of the ankle path, and angles between the shoulder, hip and ankle. Baudry et al.³ reported that the amplitudes of the double-leg circles can be evaluated by the diameters of the ankle and body alignment. Fujihara et al.¹⁴ proposed a simple measure of the amplitude based on a horizontal component of head-toe distance. Nawa et al.⁹ proposed a measure of a smooth rotation of the double-leg circles by observing the synchronization between the tip-toe circling around the vertical axis and the body-rolling around the body’s longitudinal axis. However, the pursuit of the quality can only be possible within the range where its dynamic balance is maintained.

Empirical evidence indicates that a small change in motion during pommel horse exercises can readily result in falls. This suggests the difficulty of maintaining dynamic balance during this movement. The physical laws governing the dynamic balance can have a significant impact on the possible motions in the circles. However, the theoretical background for understanding the dynamic balance is largely unclear. Very few studies have theoretically examined mechanical principles of circles. Smith¹⁵ pointed out a mechanical analogy between the dynamics of a conical pendulum and the dynamics of the CoM during the circles. Fujihara et al.^7,16 extended it to two conical pendulums with two support points. The conical pendulum models, which explain only the CoM motion, have been mostly used only for intuitive explanations where the correspondence of the dynamical variables compared to the real exercise is unclear. de Leva¹⁷ constructed a rigid-2-link model of a pommel horse exercise, analyzed its rotation around the vertical axis, and reported the motion that requires no torques at the shoulder and the wrist joints. However, the rotation of the body around its longitudinal axis has not been taken into account. Multi-body models with higher degrees of freedom are effective in quantitatively reproducing experimental data, but due to their complexity are not necessarily effective in simplifying and understanding the fundamental dynamics. After all, little progress has been made in understanding the dynamic balance of the circles with three-dimensional rotation.

The purpose of this study is to propose a simple theoretical model of the dynamic balance of the circles with rotations of the body not only around the vertical axis but also around the longitudinal axis, and to illustrate its qualitative properties. To this end, a simple model of the dynamics and a dynamic balance condition of symmetric circles with zero torques at the wrist and the shoulder joints are assumed in most analyses. A control law is proposed to demonstrate the circles. The various effects of the physical and motion parameters on the dynamic balance condition are analyzed. For more realistic situations, the analysis is also performed for circles under non-zero joint torque conditions and under asymmetric trajectory conditions. In addition, the mechanical principles of the circles are discussed regarding the pommel reaction forces, the conical pendulum model and its extension, the dynamic balance condition, and angular momentum changes.

This paper is partly based on our conference paper in Japanese¹⁸, with considerable new results including detailed analyses of the dynamic balance condition with symmetric and asymmetric motions.

Model

Physical model

We assume a physical model composed of a body segment (head, neck, trunk, and lower limbs) and an arm segment (Fig. 1). The arm segment has no weight. The body segment is one rigid body that is axisymmetric with respect to the longitudinal axis. A shoulder joint is located on the longitudinal axis of the body segment. A single point of support at a wrist joint is assumed, without distinguishing between the single-hand support and the double-hand support. The origin of the global reference frame is at the support point.

Figure 1

Simple model of circles with one support point on pommel horse exercise.

Equations of motion

The dynamics of the model is described by Newton-Euler’s motion equations (e.g,^19,20) for a rigid body moving freely in three dimensional space (Fig. 1),

$$\begin{aligned} {\left\{ \begin{array}{ll} m \ddot{\varvec{p}}_\text{G} + m \varvec{g} = \varvec{f}, \\ \varvec{I} \dot{\varvec{\omega }} + \varvec{\omega } \times ( \varvec{I} \varvec{\omega } ) = \varvec{n}, \end{array}\right. } \end{aligned}$$

(1)

where $\varvec{p}_\text{G} = [x_\text{G}, y_\text{G}, z_\text{G}]^\top$ is the position vector of the CoM, $\varvec{f}=[f_x, f_y, f_z]^\top$ is the net force vector applied to the body segment, $\varvec{\omega }$ is the angular velocity vector of the body segment, $\varvec{n}$ is the net torque vector applied to the body segment, m is the body mass, and $\varvec{g}=[0,0,g]^\top$ is the gravitational acceleration vector with $g = 9.8 \ \mathrm m/s^2$. These vectors are represented in the global reference frame fixed at the ground. The matrix $\varvec{I} = \varvec{R}(\varvec{\theta }) \bar{\varvec{I}} \varvec{R}(\varvec{\theta })^\top$ is the inertia tensor of the body segment around the CoM, represented in the global reference frame, where $\bar{\varvec{I}}=\text{diag} (\bar{I}_x, \bar{I}_y, \bar{I}_z )$ is the inertia tensor of the body segment around the CoM, represented in the local reference frame of the body segment. The matrix $\varvec{R}(\varvec{\theta }) = \varvec{R}_z(\theta _1) \varvec{R}_x(\theta _2) \varvec{R}_y(\theta _3)$ represents the orientation of the body segment with respect to the global reference frame, where $\varvec{\theta } = [\theta _1, \theta _2, \theta _3]^\top$ is the vector of Euler angles (Z-X-Y Euler angle), and $\varvec{R}_z$, $\varvec{R}_x$, and $\varvec{R}_y$ are the rotation matrices around the z, x, and y axes, respectively. The angles $\theta _1$ and $\theta _2$ represent the rotations of the body in the horizontal and vertical planes, respectively. The angle $\theta _3$ represents the rotation of the body around its longitudinal axis. The condition $\theta _1 = \theta _2 = \theta _3=0$ corresponds to the orientation in which the body’s longitudinal axis is parallel to the x-y plane, the head is toward the positive y direction.

The angular velocity $\varvec{\omega }$ and the time-derivative of the Euler angle vector $\dot{\varvec{\theta }}=[\dot{\theta }_1, \dot{\theta }_2, \dot{\theta }_3]^\top$ have the relation:

$$\begin{aligned}&\varvec{\omega } = \varvec{K} \dot{\varvec{\theta }}, \end{aligned}$$

(2)

$$\begin{aligned} \varvec{K} = \begin{bmatrix} 0 &{} \cos \theta _1 &{} -\sin \theta _1 \cos \theta _2 \\ 0 &{} \sin \theta _1 &{} \cos \theta _1 \cos \theta _2 \\ 1 &{} 0 &{} \sin \theta _2 \end{bmatrix}. \end{aligned}$$

(3)

This is derived from $\hat{\varvec{\omega }} = \dot{\varvec{R}}\varvec{R}^\top$, where $\hat{\varvec{\omega }}$ represents the skew-symmetric matrix of $3 \times 3$ converted from $\varvec{\omega }$ vector (e.g.,²⁰). Taking the time derivative of (2),

$$\begin{aligned} \dot{\varvec{\omega }} = \varvec{K} \ddot{\varvec{\theta }} + \dot{\varvec{K}} \dot{\varvec{\theta }}, \end{aligned}$$

(4)

is obtained.

Figure 2

Equivalence of force systems.

From the equivalence of force systems (e.g.,²¹) or from the simplification of a multi-body model (e.g.,^19,22) with the assumption that the arm segment has no weight, it can be shown that the set of the net force and torque $(\varvec{f}, \varvec{n})$ acting on the CoM is equivalent to the set of the applied force and torque $(\varvec{f}, \varvec{n}_\text{S}) = (\varvec{f}, \varvec{n} - \varvec{p}_\text{SG} \times \varvec{f})$ at the shoulder joint, and is also equivalent to $(\varvec{f}, \varvec{n}_\text{H}) = (\varvec{f}, \varvec{n} - \varvec{p}_\text{HG} \times \varvec{f})$ at the wrist joint, where $\varvec{p}_\text{SG}$ is the position vector from the CoM to the shoulder joint, $\varvec{p}_\text{HG}$ is the position vector from the CoM to the wrist joint, $\varvec{n}_\text{H}=[n_\text{H,x}, n_\text{H,y}, n_\text{H,z}]^\top$ is the torque vector of the wrist joint, and $\varvec{n}_\text{S}=[n_\text{S,x}, n_\text{S,y}, n_\text{S,z}]^\top$ is the torque vector of the shoulder joint (Fig. 2). Thus, we have

$$\begin{aligned} \varvec{n}&= \varvec{n}_\text{S} + \varvec{p}_\text{SG} \times \varvec{f}, \\&= \varvec{n}_\text{H} + \varvec{p}_\text{HG} \times \varvec{f}. \end{aligned}$$

(5)

By using (5), the input $(\varvec{f},\varvec{n})$ to the model (1) can be always transformed into $(\varvec{f},\varvec{n}_\text{H})$ or $(\varvec{f},\varvec{n}_\text{S})$.

Symmetric and constant velocity circles

Figure 3

Pose of symmetric circle and definitions of variables (pose at $\theta _1=0$ and $\theta _{G} = - \frac{\pi }{2}$).

We basically assume a simplified trajectory of the circle with high symmetries and constant velocities (Fig. 3), where the body rotates around the z-axis (the vertical axis) and the body’s longitudinal axis, while the arm and body segments are in the same orientation in the horizontal plane, and in the constant angles to the horizontal plane with the body angle $\theta _\text{2} = \theta _\text{2,d}$ and the arm angle $\alpha = \alpha _\text{d}$. The length $\ell _\text{HS}$ from the wrist to the shoulder joint and the length $\ell _\text{SG}$ from the shoulder joint to the CoM are constant. Owing to this symmetry, the alignment of the body segment in the vertical plane can be uniquely determined by a couple of parameters $(\theta _\text{2,d}, \alpha _\text{d})$.

Representing the CoM position $(x_\text{G}, y_\text{G})$ by $(r_\text{G}, \theta _\text{G})$ in polar coordinates, where $x_\text{G} = r_\text{G} \cos \theta _\text{G}$ and $y_\text{G} = r_\text{G} \sin \theta _\text{G}$, the motion trajectory is described as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} r_\text{G}(t) = r_\text{G,d}, \\ \theta _\text{G}(t) = \theta _1 - \frac{\pi }{2} = \omega _\text{d} t - \frac{\pi }{2}, \\ z_\text{G}(t) = z_\text{G,d}, \\ \theta _1(t) = \omega _\text{d} t, \\ \theta _2(t) = \theta _{2, \mathrm d}, \\ \theta _3(t) = -\beta \theta _1(t) \ \ = -\beta \omega _\text{d} t, \end{array}\right. } \, \, \, \, \, {\left\{ \begin{array}{ll} \dot{r}_\text{G}(t) = 0, \\ \dot{\theta }_\text{G}(t) = \omega _\text{G} = \omega _\text{d}, \\ \dot{z}_\text{G}(t) = 0, \\ \dot{\theta }_1(t) = \omega _\text{d}, \\ \dot{\theta }_2(t) = 0, \\ \dot{\theta }_3(t) = - \beta \omega _\text{d}, \end{array}\right. } \, \, \, \, \, {\left\{ \begin{array}{ll} \ddot{r}_\text{G}(t) = 0, \\ \ddot{\theta }_\text{G}(t) = 0, \\ \ddot{z}_\text{G}(t) = 0, \\ \ddot{\theta }_1(t) = 0, \\ \ddot{\theta }_2(t) = 0, \\ \ddot{\theta }_3(t) = 0. \end{array}\right. } \end{aligned}$$

(6)

The angles synchronize as to keep $\theta _\text{G} = \theta _1 - \pi /2$ and $\theta _3 = -\beta \theta _1$ with constants $\beta$. The value of $\beta$ is 1 for the double-leg circle. The angular velocities are all constant, as $\dot{\theta }_1=\omega _\text{d}$, $\dot{\theta }_2=0$, $\dot{\theta }_3=-\beta \omega _\text{d}$. The CoM of the body segment is in a uniform circular motion with the radius $r_\text{G} = r_\text{G,d}$ (constant), the angular velocity $\dot{\theta }_\text{G} = \omega _\text{d}$ (constant) around the support point, and the height $z_\text{G} = z_\text{G,d}$ (constant).

Dynamic balance condition of symmetric and constant velocity circles

If there is no limitation on joint torques, no limitation on constraint forces between the pommel and the hands, and no limitation on ability of muscular control, then any motion is possible without risk of falling. However, usually these assumptions are not true and a slight change of motion can lead to falling in pommel horse exercise. In order to maintain dynamic balance with minimum strength and with large safety margins to prevent falling against disturbances, it can be rational to select a motion with torques close to zero. Thus, inspired by de Leva¹⁷, we focus on the condition that the torques of the wrist and shoulder joints are zero:

$$\begin{aligned} \varvec{n}_\text{H}=\varvec{n}_\text{S}=0. \end{aligned}$$

(7)

which we call a dynamic balance condition in this paper. See Section 4.2 for a discussion of a rationale for this condition.

From (1), (5), and (7), the condition $\varvec{n}_\text{H} = \varvec{n}_\text{S} = 0$ in the general situation is written as

$$\begin{aligned} {\left\{ \begin{array}{ll} \varvec{n}_\text{H} = \varvec{I} \dot{\varvec{\omega}} + \varvec{\omega} \times ( \varvec{I} \varvec{\omega } ) - \varvec{p}_\text{HG} \times (m \ddot{\varvec{p}}_\text{G} + m \varvec{g} ) = 0, \\ \varvec{n}_\text{S} = \varvec{I} \dot{\varvec{\omega }} + \varvec{\omega} \times ( \varvec{I} \varvec{\omega} ) - \varvec{p}_\text{SG} \times (m \ddot{\varvec{p}}_\text{G} + m \varvec{g} ) = 0. \end{array}\right. } \end{aligned}$$

(8)

By substituting the simplified trajectory (6) at $t=0$, (8) can be reduced to the condition on $(\alpha _\text{d}, \theta _{2, \mathrm d})$ as follows (see Supplementary Information S1for details):

$$\begin{aligned} {\left\{ \begin{array}{ll} n_\text{S,x} = c_1 \cos \theta_\text{2,d} + c_2 \cos \theta_\text{2,d} \sin \theta_\text{2,d} - c_3 \cos \alpha_\text{d} \sin \theta_\text{2,d} = 0, \\ (n_\text{H,x}-n_\text{S,x} )/(m\ell_\text{HS}) = g \cos \alpha_\text{d} + c_4 \cos \alpha_\text{d} \sin \alpha_\text{d} - c_5 \sin \alpha_\text{d} \cos \theta_\text{2,d} = 0, \end{array}\right. } \end{aligned}$$

(9)

where $c_1, c_2, \ldots , c_5$ are the constants:

$$\begin{aligned}{}&c_1 = \omega_\text{d}^2 \beta ( \bar{I}_x + \bar{I}_y - \bar{I}_z) - m \ell_\text{SG} g, \end{aligned}$$

(10)

$$\begin{aligned}{}&c_2 = \omega _\text{d}^2 (\bar{I}_z - \bar{I}_y) + m \ell ^2_\text{SG} \omega _\text{d}^2, \end{aligned}$$

(11)

$$\begin{aligned}{}&c_3 = m \ell _{SG} \ell _\text{HS} \omega _\text{d}^2 > 0, \end{aligned}$$

(12)

$$\begin{aligned}{}&c_4 = \omega _\text{d}^2 \ell _\text{HS} > 0, \end{aligned}$$

(13)

$$\begin{aligned}{}&c_5 = \omega _\text{d}^2 \ell _\text{SG} > 0. \end{aligned}$$

(14)

Thus, the symmetric and constant velocity circle under the dynamic balance condition (7) can be achieved when $( \theta _{2, \mathrm d}, \alpha _\text{d} )$ satisfies (9) in the possible range of solutions $0< \theta _\text{2, d}< \alpha _\text{d} < \frac{\pi }{2}$. For a given $\omega _\text{d}$ (and other parameters), (9) are the nonlinear simultaneous equations of $(\theta _{2, \mathrm d}, \alpha _\text{d})$, which can be solved by a numerical algorithm, e.g., ‘fsolve’ in MATLAB. Or, we can derive a necessary condition in a polynomial form as shown in Supplementary Information S2.

Control law

To demonstrate the feasibility of the symmetric circle under the dynamic balance condition, we perform a forward dynamics simulation. The control objective is to achieve symmetrical circles as defined in (6). The control law is based on the feedback linearization (e.g.,²³). See Supplementary Information S3 for details.

Result

All physical parameters of the model were estimated based on the body height H. As in Table 1, the body mass m was estimated from H and the body mass index (BMI denoted as $W_\text{BMI}$)²⁴. Since children’s BMIs tend to be smaller than those of adults, BMIs for $H<1.5$ (m) were estimated by the linear interpolation between $(H, W_\text{BMI})=(1.5, 22)$ and $(H, W_\text{BMI})=(1.3, 16)$. The segment lengths were estimated by the statistical data (Fig. 3.1 of Winter²⁵) from H. The inertial moments around the frontal axis and the anteroposterior axis were averaged for symmetric modeling, and estimated by the regression equation in Tab. 1 reconstructed from the literature^26,27. The inertial moment around the longitudinal axis was approximated using the ratio to that around the frontal axis (Figure 4.13 of Zatsiorsky²⁸).

Table 1 Definitions of physical parameters.

In the following, unless otherwise stated, the body height was set to $H = 1.60$ m, the period of a circle was $T=0.9$ s ($\omega _\text{d}=2\pi /T = 6.98$ rad/s), and the rotation ratio was $\beta =1$, as typical values. Based on these, the other parameters were estimated from Tab. 1 as $m = 56.3$ kg, $\bar{\varvec{I}} = \text{diag}(8.96, 0.747, 8.96)$ $\mathrm{kg \cdot m^2}$, $\ell _\text{HS} = 0.531$ m, $\ell _\text{SG} = 0.413$ m, $\ell _\text{GA} = 0.834$ m, $\ell _\text{GT} = 1.04$ m, and the dynamic balance condition was calculated from (9) as $\alpha _\text{d} = 59.6$ deg and $\theta _{2,\mathrm d} = 20.6$ deg.

Symmetric circle under dynamic balance condition

Figure 4

Result of forward dynamics simulation starting away from dynamic balance condition.

Figure 5

Poses on symmetric circle under dynamic balance condition with $\beta = 1$. (A) The initial pose at $t=0$. The orientations of vectors are also shown: angular velocity $\varvec{\omega }$ (blue line), forces $\varvec{f}$ (cyan lines), torque $\varvec{n}$ (magenta line), and angular momenta around the CoM and the wrist joint (green lines). (B–D) Poses during the circle from different viewpoints. The positions of the shoulder joint (cyan dot), the center-of-mass (magenta dot), and the anlke joint (green dot) are shown.

First, we show the results of forward dynamics simulations of the symmetric circle under the dynamic balance condition with $\beta = 1$. The gains of the controller were set as $K_{p}=50$ and $K_{v}=2\sqrt{50}$.

Figure 4 illustrates the result of the forward simulation. The initial state $[x_\text{G}(0),y_\text{G}(0),z_\text{G}(0),\theta_1(0),\theta_2(0),\theta_3(0),\dot{x}_\text{G}(0),\dot{y}_\text{G}(0),\dot{z}_\text{G}(0),\dot{\theta}_1(0),\dot{\theta}_2(0),\dot{\theta}_3(0)]^\top$, being away from the dynamic balance condition, was given by multiplying the state on the dynamic balance condition by 0.95. Although the motion initially required non-zero torques, it gradually converged to the dynamic balance condition satisfying $\varvec{n}_\text{H}=\varvec{n}_\text{S} = 0$. This means the validity of the calculation of the dynamic balance condition and the feasibility of the circle under the dynamic balance condition.

Figure 5 illustrates the poses and vectors of physical quantities on the circle, obtained by a simulation performed from the initial state just under the dynamic balance condition. Note the symmetry of the circle around the z-axis, where one rotation of the foot around the z-axis corresponds to one rotation of the body around its longitudinal axis. The force vectors $\varvec{f}$ represented by the cyan lines at the wrist and shoulder joints are parallel to the arm segment.

Effects of physical properties on dynamic balance condition

Figure 6

Effect of body height on dynamic balance condition.

Figure 7

Effect of arm length on dynamic balance condition.

Figure 6 illustrates the effects of gymnasts’ height on the dynamic balance condition. The body height H was varied from 1.3 m to 1.9 m. The three panels on the left side of the figure show the changes in parameters satisfying the dynamic balance condition. The stick diagrams on the right side show the poses at $t=0$, where the magenta dots denote the CoMs. As the body height H increases, the arm angle $\alpha = \alpha _\text{d}$ decreases, the radii of the ankle $r_\text{A}$ and the toes $r_\text{T}$ increase, the CoM radius $r_\text{G} = r_\text{G,d}$ slightly increases, and the CoM height $z_\text{G} = z_\text{G,d}$ increases. However, the body angle $\theta _2 = \theta _\text{2,d}$, and the ankle height $z_\text{A}$ and the toe height $z_\text{T}$ show small and non-monotonic changes. It should be noted that such small changes were sensitive to the estimation of inertia parameters.

Figure 7 illustrates the effects of the arm length on the dynamic balance condition. The format of the figure is almost the same as Fig. 6. The arm length was varied by $\pm 10$ % around $\ell _\text{HS} = 0.531$ m, while the height was fixed at $H = 1.6$ m. Although the pose of the dynamic balance condition does not vary much with the arm length, the height of the toes and the CoM gradually increase with the arm length. This suggests that a gymnast with long arms may have an advantage in keeping the CoM and the legs high above the pommel horse.

Effects of period on dynamic balance condition

Figure 8

Effect of period T of one circle on dynamic balance condition.

Figure 8 illustrates the effects of the period T of a circle on the dynamic balance condition. The format is almost the same as Fig. 6. The period was varied from $T=0.7$ s to $T=1.2$ s. As the period T increases, the arm angle $\alpha _\text{d}$ and the body angle $\theta _\text{2, d}$ increase, the toe radius $r_\text{T}$ decreases, the CoM radius $r_\text{G}$ increases, and the toe height $z_\text{T}$ and the CoM height $z_\text{G}$ decrease. This suggests that the period should be shorter to get larger amplitude of the toes and to get higher position of the CoM and the toes. However, the short periods lead to the slightly small amplitude of the CoM in the horizontal plane.

In experimental observations, many findings related to the period of a circle have been reported. The velocity of the CoM was greater in high-scored gymnasts, except in the rear support phase¹⁶. The period of a circle with high CoM positions was shorter than that of a normal circle⁸. In cross-circles requiring gymnasts to raise their legs in the rear support phase, the duration of the rear support phase and the duration of a circle were shorter compared to side-circles⁷. In Russian wendeswing, the period was shorter on the floor than that on the pommel horse²⁹. These findings may qualitatively support our results that increasing the speed of the circle has the effect of lifting the body and avoiding collisions with the pommel horse.

Effects of body rotation around the longitudinal axis on dynamic balance condition

Figure 9

Effect of body rotation around longitudinal axis on dynamic balance condition. The conditions $\beta =2$, $\beta =1$ and $\beta =0$ correspond in a broad sense to the Magyar spindle, the double-leg circle, and the Russian wendeswing, respectively.

The body rotation of our model is represented by three components: the body’s rotation $\theta _1$ around the vertical axis, the body’s rotation $\theta _3$ around the longitudinal axis, and the CoM’s rotation $\theta _\text{G}$ around the vertical axis (Fig. 1). The ratio of these rotation speeds is $\dot{\theta }_1: \dot{\theta }_3: \dot{\theta }_\text{G} = 1: \beta : 1$. Here, we focus on the effect of $\beta$, i.e., the rotation around the longitudinal axis. For example, the case $\beta =1$ means the direction of the rotation as in the double-leg circle. The case $\beta =2$ corresponds to the rotation as in the Magyar spindle, where the body rotates around its longitudinal axis at a double speed compared to the double-leg circle. The case $\beta =0$ corresponds to the Russian wendeswing, in which the body does not rotate around its longitudinal axis. The cases $\beta =-1$ and $\beta =-2$ are virtual motions, in which the body rotates around its longitudinal axis in the opposite direction.

Figure 9 illustrates the effects of $\beta$. The format is almost the same as Fig. 6. The result shows that as $\beta$ increases, the arm angle $\alpha _\text{d}$ and the body angle $\theta _\text{2,d}$ decrease, and the radius $r_\text{G,d}$ and the height $z_\text{G, d}$ of the CoM increase. This suggests that the body rotation corresponding to $\beta > 0$ has the effect of lifting the body for the dynamic balance condition.

The motion without the rotation around the longitudinal axis ($\beta =0$) have been simulated by de Leva¹⁷, where the torque-free condition was $\alpha _\text{d} = 90 - 31 = 59.0$ deg and $\theta _\text{2,d} = 90 - 66.5 = 23.5$ deg with $H=1.67$ m, $m=61.6$ kg, and $T=2\pi /7.0=0.898$ s. Those are very similar to our result under the corresponding condition: $\alpha _\text{d} = 60.1$ deg and $\theta _\text{2,d} = 24.7$ deg, with $\beta =0$, $H=1.67$ m, and $T=0.898$ s.

Effects of shoulder torque and wrist torque

Figure 10

Effect of shoulder torque $n_{{{\text{S0}}}}$.

Figure 11

Effect of wrist torque $n_{{{\text{H0}}}}$.

In this subsection, we consider the relaxation of the dynamic balance condition by allowing non-zero torques of the shoulder joint (Fig. 10) or the wrist joint (Fig. 11). In Figs.  10 and 11, $\varvec{n}_\text{S}$ and $\varvec{n}_\text{H}$ were set to have a constant amplitude $n_\text{S0}$ and $n_\text{S0}$, respectively, while the other torque was set to zero. The right-hand sides of (9) were changed to $n_\text{H0}$ and $(n_\text{H0} - n_\text{S0})/( m \ell _\text{HS})$, respectively. The negative $n_\text{S0}$ and $n_\text{H0}$ act as a torque that rotates the body and arm segment in clockwise direction in Figs.  10 and 11, respectively.

Figure 10 illustrates the effects of the shoulder torque. As the shoulder torque increases, the arm angle $\alpha _\text{d}$ and the body angle $\theta _\text{2,d}$ increase, the CoM radius $r_\text{G}$ increase, and the CoM height $z_\text{G}$ decreases. This suggests that the high CoM position and the large toe amplitude can be achieved by the shoulder torque of the negative direction. Note that these effects on the poses are qualitatively analogous to those of the period (Fig. 8).

Figure 11 illustrates the effects of the wrist torque. As the wrist torque increases, the arm angle $\alpha _\text{d}$ decreases, the body angle $\theta _\text{2,d}$ increases, and the radius $r_\text{T}$, $r_\text{G}$, and the height $z_\text{T}$ decrease. The effects of the wrist torque are very small compared to those of the shoulder torque shown in Fig. 10. This suggests that the relatively weak effect of the wrist torque on the dynamic balance condition.

Note that by using the joint torques, the pose at the dynamic balance condition can be deviated in the directions and in the ranges almost shown in Figs. 10 and 11. The shoulder torque and the wrist torque can change the pose in the different directions. The ability to change poses is supposed to be greater for the shoulder torque and less for the wrist torque, partly because of the strength of the joints. For example, mean values of maximum torques in shoulder flexion/extension under isokinetic conditions have been reported to range from 48.65 ($= 97.3 / 2$) Nm to 133.4 ($=266.77 / 2$) Nm in male gymnasts, and ranged from 37.84 ($=75.67 / 2$) to 112.6 ($=225.22 / 2$) in male non-gymnasts³⁰; mean values of maximum torques in wrist flexion/extension under isometric conditions have been reported to ranged from 6.6 to 11.9 Nm in males³¹. Therefore, it is suggested that the range of the modification of the dynamic balance condition by the joint torques is restricted in this sense.

Effect of asymmetric motion

Figure 12

Effect of relative angular velocity of CoM $\dot{\theta }_{{\text{G}}} = \gamma \omega _{{\text{d}}}$ on dynamic balance condition.

Figure 13

Effect of vertical acceleration of CoM $\ddot{z}_{{\text{G}}}$ on dynamic balance condition.

While in the previous subsections we analyzed the highly symmetric motion with constant velocities, in experimental observations various asymmetries with fluctuating velocities are reported. Here, we present two examples of many factors associated with the asymmetric rotations. Note that, due to the asymmetry, the analysis here focuses only on the state at a certain instant of time during the circle. This state of balance is only possible at this instant. It is not possible to keep rotating while maintaining the same pose, unlike the previous subsections. The dynamic balance condition in this section were calculated based on (S1.4) in Supplementary Information S1.

Figure 12 shows the effect of slowing down the angular velocity $\dot{\theta }_\text{G}$ of the CoM around the support point with $\dot{\theta }_\text{G}=\gamma \omega _\text{d}$, $0 < \gamma \le 1$, and $\omega _\text{d}=2\pi /0.9$ rad/s. The ratio of the rotation speeds was $\dot{\theta }_1: \dot{\theta }_3: \dot{\theta }_\text{G} = 1: 1: \gamma$. The result shows that as $\gamma$ decreases, i.e., as $\dot{\theta }_\text{G}$ decreases, the radius $r_\text{G}$ increases, up to $r_\text{G} > 0.30$ m when $\gamma =0.2$. This radius is larger than the results in the previous subsections, which are in the range $r_\text{G} < 0.2$ m. This suggests that in this asymmetric motion, the large amplitude of the CoM can be achieved by the small angular velocity of the CoM in the horizontal plane.

In experimental observations, the CoM trajectory of the circles on two handles of the pommel horse has been reported to be elliptical, and be shaped like two arcs centered at two support points in the horizontal plane⁵. The CoM trajectory became more circular during cross-circles with narrower support-hand distances⁷, and during the Flop motion on one handle¹⁰. Thus, the CoM trajectory apppears to be affected by the distance of the support points. For the angular velocity of the CoM, it is worth noting that, in the circles on one support point, the CoM should make one rotation around the support point in the horizontal plane, while the body makes one rotation around the vertical axis, i.e., $\dot{\theta }_\text{G} \simeq \dot{\theta }_1$. In contrast, in the circles on two handles, the CoM should not make one rotation around either the left or the right handle, while the body makes one rotation around the vertical axis, i.e., $\dot{\theta }_\text{G} < \dot{\theta }_1$. An example of $\gamma$ visually read from Fig.  6 in Fujihara et al.⁵ are almost $\gamma \simeq 45/93 = 0.48$ in the frank-in phase, where the CoM rotates 45 deg around the support point while the body rotates 93 deg in the horizontal plane. For the radius of the CoM, a relatively small value of $r_\text{G}$ was observed in the Flop motion on one handle¹⁰, where the range was about $r_\text{G} < 0.2$ m. A relatively large value of $r_\text{G}$ was observed in the single-hand support phase of the circles on two handles⁵, where $r_\text{G}$ was about $r_\text{G} \simeq 0.30$ m. These facts are roughly consistent with our analysis (Fig.  12) that $r_\text{G}$ increases as $\gamma$ decreases.

Figure 13 shows the effect of the vertical acceleration $\ddot{z}_\text{G}$ of the CoM with $\gamma =0.5$. In the CoM motion, $\dot{z}_\text{G}=0$ and $\ddot{z}_\text{G} \ne 0$ were assumed, while in the previous analysis a constant height of CoM, i.e., $\dot{z}_\text{G}=\ddot{z}_\text{G}=0$, was assumed. The result shows that as $\ddot{z}_\text{G}$ increases, the body angle $\theta _2$ increases, and the toes drop down. Note that, owing to the motion equation $m \ddot{z}_\text{G} + m g = f_z$, the CoM acceleration $\ddot{z}_\text{G}$ is directly related to the vertical pommel reaction force $f_z$. Thus, a positive $\ddot{z}_\text{G}$ means weighting, because the pommel reaction force is greater than the body weight, i.e. $f_z > mg$. Conversely, a negative $\ddot{z}_\text{G}$ means unweighting, i.e. $f_z < mg$. These suggest that in this asymmetric motion, the weighting with upward CoM acceleration is related to the low CoM position.

In experimental observations of the circles on two handles, the CoM moves vertically as well as horizontally, and especially in a rear support phase, the CoM height $z_\text{G}$ is the lowest (i.e., $\ddot{z}_\text{G}$ would be positive), the vertical component $f_z$ of the pommel reaction force is large (i.e., weighting on the handle), and the toes are lower⁵. This fact is qualitatively consistent with the result shown in Fig. 13 that $z_\text{T}$ decreases as $\ddot{z}_\text{G}$ increases.

The dynamic balance condition is generally described by nonlinear equations with various parameters (e.g., constants, positions, velocities, and accelerations). It is clear from this paper that the dynamic balance of the circles is influenced by a variety of factors. Therefore, in addition to the examples shown above, many other factors can affect the dynamic balance. It should be noted that the relative position of the body segment and the arm segment, i.e., the position of the shoulder joint, affects the dynamic balance (see Fig. 4.1 of Supplementary Information S4). If detailed measured data are available, these conditions of the simple model can be adjusted closer to the measured data. Such examples are shown in Supplementary Information S4 for readers who wish to analyze asymmetric motions.

Mechanical understanding of circles

Figure 14

Physical variables of symmetric circle on dynamic balance condition with $\beta =1$ at $t=0$. (A) Forces exerted on the body segment and the arm segment. (B) Conical pendulum interpretation of the translational motion of the CoM. (C) Change of angular momentum around the CoM. (D) Change of angular momentum around the hand joint.

Pommel reaction force vector

Under the dynamic balance condition the pommel reaction force vector $\varvec{f}$ must be parallel to the arm segment in the simple model (Fig. 5 and Fig. 14A ), due to the approximation that the arm has no weight. This can be confirmed by the relationship obtained from (5),

$$\begin{aligned} \varvec{n}_\text{H} - \varvec{n}_\text{S} = (\varvec{p}_\text{SG}-\varvec{p}_\text{HG}) \times \varvec{f}. \end{aligned}$$

This means that when the torques at the wrist and shoulder joints are equal (or zero), i.e., $\varvec{n}_\text{H} = \varvec{n}_\text{S}$ (or $\varvec{n}_\text{H} = \varvec{n}_\text{S}=0$), the pommel reaction force is directed along the arm, i.e., $(\varvec{p}_\text{SG}-\varvec{p}_\text{HG}) \times \varvec{f}=0$. Interestingly, a similar situation has also been observed experimentally⁵, in which the hand force vector is almost directed toward the shoulder joint.

Dynamic balance condition

One may think that it is questionable for the condition on muscles to keep intersegmental torques strictly at zero all the time, while exerting the necessary amount of intersegmental forces during a complex body movement in complex gymnastics routines. In fact, Yamada et al.^13,32 estimated non-zero joint torques during circles based on inverse dynamics, where the estimated shoulder torque fluctuated and the absolute values of the torque components on the supporting side were less than 100 Nm for most of the time¹³. They claimed that the shoulder muscles do not act as a driving force, and that the shoulder joint may contribute as a fixed fulcrum. Thus, it would not be realistic to assume that the joint torques are never used in the circles.

Nevertheless, there is some rationale for selecting a motion with torques close to zero, in order to maintain the dynamic balance with minimal strength and large safety margins to prevent falling. In addition, the following aspects may support that the actual motion is around the dynamic balance condition: (1) It is observed in the experiment⁵ that the pommel reaction force is roughly directed along the arm segment. (2)When the pommel reaction force is directed along the arm segment, the torques at the wrist and shoulder joints must be equal in the simple model, as discussed above. (3) The magnitudes of the joint torques are limited. In particular, the wrist joint is relatively weak, as discussed in Section 2.5. (4) In circles without grasping the handle, the wrist joint of the model can be roughly interpreted as the center of pressure (CoP) on the palm contact surface in the single-hand support phase. The CoP is also referred to as the zero moment point (ZMP), where the torque components parallel to the contact plane are zero³³.

Thus, even if the dynamic balance condition may not be strictly achieved, it can be meaningful to understand the actual motion by focusing on a fluctuation around or a deviation from this condition. As shown in Section 2.5, the joint torques can change the pose within a range centered on the dynamic balance condition. In this sense, the dynamic balance condition can be a key to understanding the dynamic balance of the circles.

CoM motion and conical pendulum

The circular motion of the CoM during the circles has been explained by the analogy to the conical pendulum^15,16. Our simple model with the symmetric motion under the dynamic balance condition also includes the conical pendulum, where the CoM makes a uniform circular motion around the z- axis in the horizontal plane (Fig. 14B). The centripetal force yielding the circular motion is the horizontal component of the inter-segmental force acting on the shoulder joint $f_y=|\varvec{f}|\cos \alpha _\text{d}$, since $\varvec{f}$ is parallel to the arm segment. The horizontal force has the relation $f_y = m r_\text{G} \omega _\text{G}^2$ with the angular velocity $\omega _\text{G}$ and the the radius of rotation $r_\text{G}$ around the vertical (z) axis. The vertical force has the relation $f_z = m g = |\varvec{f}| \sin \alpha _\text{d}$. By eliminating $|\varvec{f}|$ from these relations, we get the condition

$$\begin{aligned} g&=r_\text{G} \omega _\text{G}^2 \tan \alpha _\text{d}. \end{aligned}$$

(15)

This condition (15) is also derived from (9), and the geometrical relations ((S1.6) and (S1.7), i.e., (15) is a necessary condition but not a sufficient condition for the dynamic balance condition (9).

Let us compare the condition (15) in the conical pendulum model with the measured data of a normal circle on two handles. In the left-hand support phase of a circle on two handles, the approximate variables can be visually read from the literature⁵ as $\alpha _\text{d} = 72 \ \text{deg} =1.26 \ \text{rad}$, $r_\text{G}=0.30$ m, $\omega _\text{G}= \dot{\theta }_\text{G} =155 \ \mathrm{deg/s} = 2.71 \ \mathrm{rad/s}$ (see Supplementary Information, Table. S4.1), then the right hand side of (15) is estimated as $r_\text{G} \dot{\theta }_\text{G}^2 \tan \alpha = 6.76 \ \mathrm{m/s^2}$. However, this is smaller than the left-hand side $g = 9.8 \ \mathrm{m/s^2}$, i.e., the estimation error is $3.04 \ \mathrm{m/s^2}$. The conical pendulum model, assuming a symmetric trajectory, might be difficult to explain this situation in a quantitative sense.

One of the reasons for this error may be due to the asymmetric rotation associated with the non-zero value of the CoM acceleration. In fact, if the accelerations of the CoM ($\ddot{z}_\text{G}$ and $\ddot{r}_\text{G}$) are not assumed to be zero, the above equation (15) is modified as

$$\begin{aligned} g + \ddot{z}_\text{G} =(r_\text{G} {\omega }_\text{G}^2 - \ddot{r}_\text{G}) \tan \alpha _\text{d}, \end{aligned}$$

(16)

(see the Supplementary Information S1 for its derivation). The vertical pommel reaction force in the left-hand support phase is about $f_z = 0.7 m g$ N in the literature⁵. The value of $\ddot{z}_\text{G}$ is estimated to be $(0.7-1)g = -2.94 \ \mathrm{m/s^2}$ from the equations of motion $\ddot{z}_\text{G} = f_z/m - g$. If the trajectory of the CoM in the horizontal plane is an arc centerd at the support point, $\ddot{r}_\text{G}$ can be assumed to be zero. The left-hand side of (16) is calculated as $9.8 + (-2.94) = 6.86 \ \mathrm{m/s^2}$, which is close to the right-hand side $6.76 \ \mathrm{m/s^2}$, where the error $0.1 \ \mathrm{m/s^2}$ is small. On the other hand, based on the data in the right-hand suppport phase of the literature⁵, the left-hand side of (16) is calculated as $9.8 + (-1.27) = 8.53 \ \mathrm{m/s^2}$, the right-hand side is $6.50 \ \mathrm{m/s^2}$, and the error $2.03 \ \mathrm{m/s^2}$ is not small. However, this error can be eliminated by assuming $\ddot{r}_\text{G}=-0.576$ $\mathrm m/s^2$, while other factors, such as non-zero joint torques, may also be responsible for this error. Thus, a more detailed consideration of the asymmetric trajectory may provide a more quantitative explanation of the CoM motion observed in realistic situations.

Change of the angular momentum

How can a gymnast increase the speed of a circle during motion? As a clue to this question, let us consider how to change the angular momentum of the body rotation around the support point (the wrist joint). We recall that the Euler’s equation (the second equation of (1)) is described by the time derivative of the angular momentum vector $\varvec{L}_\text{G} = \varvec{I}\varvec{\omega }$ around the CoM, as

$$\begin{aligned} \dot{\varvec{L}}_\text{G} = \varvec{I} \dot{\varvec{\omega }} + \varvec{\omega } \times ( \varvec{I} \varvec{\omega }) = \varvec{n}. \end{aligned}$$

(17)

The angular momentum vector around the wrist joint is written as $\varvec{L}_\text{H} = \varvec{L}_\text{G} + \varvec{p}_\text{G} \times m \dot{\varvec{p}}_\text{G}$, and the time derivative of the $\varvec{L}_\text{H}$ is written as

$$\begin{aligned} \dot{\varvec{L}}_\text{H}&= \dot{\varvec{L}}_\text{G} + \varvec{p}_\text{G} \times m \ddot{\varvec{p}}_\text{G} \\&= \varvec{n} + \varvec{p}_\text{G} \times \varvec{f} + \varvec{p}_\text{G} \times (-m \varvec{g}) \\&= \varvec{n}_\text{H} + \varvec{p}_\text{G} \times (-m \varvec{g}), \end{aligned}$$

(18)

where we used $m\ddot{\varvec{p}}_\text{G}= \varvec{f} - m \varvec{g}$ from (1) and $\varvec{n}_\text{H} = \varvec{n} + \varvec{p}_\text{G} \times \varvec{f}$ from (5). The z component of the term $\varvec{p}_\text{G} \times (-m \varvec{g})$ is zero, because the x and y components of the vector $\varvec{g}$ are zero. Thus, as long as the z component of $\varvec{n}_\text{H}$ is zero, the angular momentum of the body around the support point cannot change its z component. In other words, the z component of the hand torque $\varvec{n}_\text{H}$ is needed to change the z component of angular momentum of the body supported at one point. Note that this property holds regardless of the shoulder torque, and that this property also holds for a model composed of multiple segments on one support point.

The change of x and y components of the angular momentum vector $\varvec{L}_\text{H}$ is considered as follows. On the symmetric circles at $t=0$ ($\theta _1 = 0$, $\theta _\text{G} = -\pi /2$), the body is in the position orthogonal to the x-axis as shown in Fig.  5A and Fig.  14, in which $\varvec{p}_\text{G}$, $\varvec{\omega }$, $\varvec{L}_\text{G}$, and $\varvec{L}_\text{H}$ are all in the y-z plane. The angular velocity vector $\varvec{\omega }$, shown by the blue line in Figs.  5A and 14C and D, is composed of two rotations with the same speed: rotations around the vertical axis and the body’s longitudinal axis. The vector $\varvec{L}_\text{H}$, shown by the green lines in these figures, varies according to $\dot{\varvec{L}}_\text{H} = \varvec{p}_\text{G} \times (- m \varvec{g} )$ when $\varvec{n}_\text{H}=0$. Due to the symmetry of the trajectory, $\varvec{p}_\text{G}$, $\varvec{L}_\text{H}$, and $\varvec{g}$ are always on the same vertical plane, the torque by the gravity, $\varvec{p}_\text{G} \times (- m \varvec{g} )$, does not change the amplitude of $\varvec{L}_\text{H}$, but always changes the direction of $\varvec{L}_\text{H}$ toward the direction orthogonal to $\varvec{L}_\text{H}$ and the z axis. Therefore, the vector $\varvec{L}_\text{H}$ continues a precession, drawing a cone around the z axis. Denote the constant amplitude of the xy component of $\varvec{L}_\text{H}$ by $L_{\text{H},xy}$, the x component of $\varvec{L}_\text{H}$ by $L_{\text{H},x} = L_{\text{H},xy}\cos \theta _\text{G}$, and the y component of $\varvec{P}_\text{G}$ by $P_{\text{G},y} = r_\text{G,d}\sin \theta _\text{G}$. Then, the x component of $\dot{\varvec{L}}_\text{H} = \varvec{p}_\text{G} \times (- m \varvec{g} )$ can be written as

$$\begin{aligned} \dot{L}_{\text{H},x}&= -L_{\text{H},xy} (\sin \theta _\text{G}) \dot{\theta }_\text{G} \\&= -mg r_\text{G,d}\sin \theta _\text{G}. \end{aligned}$$

This yields the simple relation between the angular momentum and the angular velocity:

$$\begin{aligned} \dot{\theta }_\text{G} =\omega _\text{d} =\frac{ mg r_\text{G,d} }{L_{\text{H},xy} }. \end{aligned}$$

(19)

Above mechanics can be compared to that of a spinning top with precession (e.g.,³⁴). Their mechanics are partly similar, but the composition of the inertia tensor and the relations among the CoM position $\varvec{p}_\text{G}$, the angular velocity $\varvec{\omega }$, and the angular momentum $\varvec{L}_\text{H}$ are very different. Furthermore, the motion of the circles would have to be stabilized by a neural controller to be realized, unlike the motion of the top.

Implications and limitations

The pommel horse exercise has complex dynamics that is very different from that of other exercises. It is not at all obvious how to simplify and qualitatively understand its fundamental dynamics. Hence, an idealized model is needed to serve as a reference for understanding the dynamics of the pommel horse exercise. We proposed the simple theoretical model of the dynamic balance of the circles with the symmetric body and the symmetric motion. The mass of the arm, which almost corresponds to 5 % (one hand) or 10 % (both hands) of the body weight²⁵, was ignored. The idea of the mass-less arm helped to simplify many analyses. It allowed to consider symmetrical motion without distinguishing between one-handed and two-handed support, allowed to the clear analysis of the direction of the pommel reaction force (Section 4.1), and allowed to change the position of the shoulder joint relative to the body segment (Supplementary Information S4).

Historically, the double-leg circle has been developed from a bent-hip (pike) position to an extended-hip (layout) position³⁵. The current code of points of the International Gymnastics Federation (³⁶ p.54) states that ‘strength and hold elements are not permitted,’ and ‘ideally, circles and flairs must be performed with complete extension.’ Actually, most studies for evaluating circles (e.g.,^3,14,9) regard symmetric motion with extended position as highly skilled. Thus, a highly skilled circle might be regarded as the motion that requires minimum strength, the extended position, and the symmetry in a broad sence. In this context, the symmetric condition of our analysis might be a distant extension of the historical development of the double-leg circles, even if it cannot be realized.

Various aspects of asymmetry have been observed in experiments, such as the postural flexion³, the elliptical orbits of motion⁵, the fluctuations of rotation speeds^5,9, the angular momentum³⁷, and the muscle forces¹³. For example, the CoM velocity is very slow during the double-hand support phase⁵. The application of the joint torques can influence the symmetry of the trajectories. Other factors, such as the physical characteristics of the body, the range of motion of joints, the force exertion characteristics of the muscles, and so on, can be also responsible for the asymmetry. The asymmetry seen in circles on two support points would be partly due to the risk of collision with the horse and the distance between the support points⁷. In this context, the symmetrical motion analyzed here may be better compared, rather than a circle on two pommels, to a circle on a single handle or on an implement (e.g., ‘mushroom’) without the risk of collision with a pommel horse.

However, prior to understanding complex and asymmetric motions, one must understand simpler and symmetric motions. It should be noted that even the fundamental dynamics of the symmetric circle was not well understood before. In fact, the conventional models of circles for understanding the dynamic balance have had a large gap with actual circles. In this paper, we focused on the symmetric circle, and showed concretely the mechanical relationship between the conical pendulum¹⁵ and the simple model of the circles (Section 4.3). We analyzed the dynamic balance with the body’s rotations around not only the vertical axis but also the longitudinal axis, which had not been studied in the conventional study¹⁷, and showed that the rotation of the body around its longitudinal axis in the double-leg circle has the effect of lifting the toes. (Section 3.4). We also illustrated that the dynamic balance of the circles is influenced by the various factors (Figs. 6,7, 8, 9, 10, 11, 12 and 13). These results provide some insight into understanding and improving the pommel horse exercise. For example, we showed that the circles with high CoM positions and large ankle amplitudes were predicted to be achieved in a fast circle (with short period of a circle), which was qualitatively consistent with the experimental observations (Section 3.3). Furthermore, we showed that a careful comparison would also provide a better understanding of asymmetric circles. The large radius of the rotation of the CoM around the support point was predicted on the conditions with the smaller angular velocity of the CoM relative to the angular velocity of the body rotation around the CoM (Section 3.6). The lower position of the toes during asymmetric circles was predicted on the conditions with weighting on the handle due to the vertical movement of the CoM. These were qualitatively consistent with the experimental observations (Section 3.6). Even the poses experimentally observed in the circles on two pommels could be analyzed by the careful analysis assuming asymmetric rotations (Supplementary Information S4).

Using a multi-body model with more degrees of freedom is a reasonable approach to quantitatively reproduce asymmetric circles. However, it is difficult for the multi-body model with both arms to simulate a perfectly symmetric circle, as in this paper, because its body structure is asymmetric. In addition, models with large degrees of freedom have a large amount of redundancy in the motion trajectories (Bernstein’s problem³⁸), which requires additional assumptions for trajectory generation on a virtual condition (e.g., a cost function in an optimization problem) and extensive numerical computations. Thus, it is not necessarily suitable for the simplified understanding of the symmetric circles under various hypothetical conditions.

Conclusion

The purpose of this study was to propose the simple theoretical model of the dynamic balance of the circles, and to illustrate its various qualitative properties. To this end, the body of a gymnast was simply modeled as one rigid body with one support point, and symmetric and constant-velocity circles are assumed in most analyses. The condition that the torques of wrist and shoulder joints are zero was assumed as a dynamic balance condition with minimum strength. A control law was proposed to demonstrate the motion. Various effects on the dynamic balance condition were analyzed, focusing on physical conditions, kinematic conditions, and conditions associated with asymmetric rotation.

The main results were as follows. In the symmetric circles, (1) as the period of a circle decrease, the heights and the radius of the toes increase, and (2) the rotation of the body around its longitudinal axis in the double-leg circle has the effect of lifting the toes. (3) The shoulder and wrist torques can change the pose around the dynamic balance condition. In asymmetric circles, (4) the radius of the CoM increases as the angular velocity of the CoM around the support point decreases, and (5) the body angle with respect to the horizontal plane increases as the upward acceleration of the CoM increases.

Moreover, mechanical principles of the circles were discussed as follows. (6) The CoM motion of the proposed model is physically related to the conical pendulum model, which needs to be extended to explain the asymmetric circles. In the simple model, (7) the pommel reaction force should be parallel to the arm segment when the wrist torque and the shoulder torque are equal or zero, and (8) to change the vertical component of the angular momentum of the body around the support point the wrist torque around the vertical axis is needed.

The previous studies on the dynamic balance have been limited to the analogy between the CoM motion and a conical pendulum^15,16 and to the theoretical model without considering the rotation of the longitudinal axis¹⁷. The dynamic balance of the circle with three dimensional rotation of the body including the rotation around the longitudinal (craniocaudal) axis was clarified for the first time in this study. These results provide theoretical and qualitative insights into understanding and improving pommel horse exercise.