Three-dimensional kinematic analysis of the golf swing using instantaneous screw axis theory, part 1: methodology and verification

Abstract

A number of recent studies have measured the extent and timing of segment rotation during the golf swing. A promising technique, instantaneous screw axis (ISA) theory, could provide a better expression of segment rotation. In Part 1 of this two-part study, the objectives are to identify the ISA of the pelvis, shoulders and left arm during the downswing, compute segment angular velocity relative to that segment’s ISA and verify that ISA theory is a valid tool to analyse segment rotation during the golf swing. Results indicate that for all subjects, at least 71% of marker velocity is a result of rotation about their respective ISA, when averaging results over the duration of the downswing, confirming that motion is primarily rotational. Furthermore, ISA position and orientation of each segment approaches, on average, the expected gross axis of rotation, confirming that motion about the ISA is representative of joint motion.

Appendices

Appendix 1: Instantaneous screw axis (ISA) computation

Position and orientation of a RB ISA can be computed from the displacement of three, non-collinear, features belonging to that body [36]. To illustrate RB ISA computation, three non-collinear markers M1, M2 and M3, were positioned at the outer edge of the cylinder, at a radius r from the cylinder centroidal axis (Fig. 9).

Fig. 9

Analytical cylinder; a isometric view, b xy-plane projection

To ensure non-collinear markers, three phase angles θ ₁, θ ₂ and θ ₃ (θ ₁ = 0), were introduced between markers, expressed relative to the positive x-axis. For simplicity, a single screw motion was imposed on the cylinder, consisting of a constant angular velocity $\vec{\omega }$ about the centroidal axis and a constant translation of velocity $\vec{v}_{//}$, parallel to the centroidal axis (Fig. 10).

Fig. 10

Analytical cylinder; a applied screw motion, b marker velocities

From the imposed motion of the cylinder, the position of each marker, relative to the global reference frame, is given by

$$ \vec{p}_{i} = x_{i} (t)\vec{i} + y_{i} (t)\vec{j} + z_{i} (t)\vec{k} $$

(1)

where x _i (t) = r cos (ωt + θ _i ); y _i (t) = r sin (ωt + θ _i ); z _i =  v _//  t and where i = [1, 2, 3], for each of the three markers, and t is the time variable. The velocity of each marker consists of a component parallel, $\vec{v}_{//}$, and perpendicular, $\vec{v}_{ \bot }$, to ISA (Fig. 10). The parallel component is equal to RB parallel velocity, v _// . The perpendicular component is calculated as the cross product of the applied angular velocity, $\vec{\omega }$, and the radius of the markers relative to the origin. The total velocity of each marker is thus given by

$$ \begin{aligned} \vec{v}_{i} (t) & = \vec{v}_{{ \bot_{i} }} + \vec{v}_{//} \\ & = [\omega \vec{k} \times (x_{i} (t)\vec{i} + y_{i} (t)\vec{j} + z_{i} (t)\vec{k})] + v_{//} \vec{k} \\ \end{aligned} $$

(2)

where $\vec{\omega }$ and $\vec{v}_{//}$ are both vectors in the z-axis direction, coinciding with the centroidal axis of the cylinder (Fig. 10).

From marker displacements RB ISA is computed in two steps, determine first the orientation of a unit vector parallel to the ISA, labelled $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$, and second the location of the ISA relative to the global reference frame. $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ is determined by computing the cross product of the relative velocity of marker M2 to marker M1, $\overrightarrow{{v}_{2 - 1}}$ and the relative velocity of marker M3 to marker M1, $\overrightarrow{{v}_{3 - 1}}$ [36]:

$$ \begin{aligned} \overrightarrow {{{\text{ISA}}_{\text{o}} }} & = {\frac{\overrightarrow{{v}_{2 - 1}} \times \overrightarrow{{v}_{3 - 1}} }{{\left\| {\overrightarrow{{v}_{2 - 1}} \times \overrightarrow{{v}_{3 - 1}}} \right\|}}} \\ & = {\frac{{\left( {\vec{v}_{2} - \vec{v}_{1} } \right) \times \left( {\vec{v}_{3} - \vec{v}_{1} } \right)}}{{\left\| {\left( {\vec{v}_{2} - \vec{v}_{1} } \right) \times \left( {\vec{v}_{3} - \vec{v}_{1} } \right)} \right\|}}} \\ \end{aligned} $$

(3)

where v ₁, v ₂ and v ₃ represent the velocity of markers M1, M2 and M3, respectively, and computed from Eq. (2). Note that the choice of marker M1 is arbitrary and could be chosen as any of the other two markers. In the present case, the parallel component of marker velocity, $\vec{v}_{{//_{i} }}$, is consistent for all three markers and the perpendicular component of marker velocity is contained within the xy plane. Therefore, the relative velocities $\overrightarrow{{v}_{2 - 1}}$ and $\overrightarrow{{v}_{3 - 1}}$ are both contained within the xy plane, and their cross product will produce a vector in the $\overrightarrow {k}$ direction. Therefore, $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ will be parallel to the applied angular velocity $\vec{\omega }$, consistent with a screw motion. Orientation of $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ with respect to the parallel vector component of each marker, $\vec{v}_{{//_{i} }}$, must be determined as the cross product cannot determine the direction of parallel velocity. Dot product between the velocity of marker M1 and $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ is computed, ISA_{Diagnostic 1}, as follows:

$$ \begin{aligned} {\text{ISA}}_{\text{Diagnostic 1}} & = \overrightarrow {{{\text{ISA}}_{\text{o}} }} \cdot \vec{v}_{1} \\ & = \overrightarrow {{{\text{ISA}}_{\text{o}} }} \cdot (\vec{v}_{{ \bot_{1} }} + \vec{v}_{{//_{i} }} ) \\ & = \overrightarrow {{{\text{ISA}}_{\text{o}} }} \cdot \vec{v}_{{//_{i} }} \\ \end{aligned} $$

(4)

where, in the present case, the perpendicular component $\vec{v}_{{ \bot_{i} }}$ is dropped as its dot product with $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ is zero since they are perpendicular vectors. Furthermore, the magnitude of ISA_{Diagnostic 1} is equal to $\left| {\vec{v}_{{//_{1} }} } \right|$, as the two vectors are parallel. If ISA_{Diagnostic 1} is positive, then $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ and the parallel velocity of all markers are in the same direction. However, if the value of ISA_{Diagnostic 1} is negative, then $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ and the parallel velocity component of all markers are of opposite direction. With $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ direction established, parallel and perpendicular components of marker velocity (Fig. 10) are computed as

$$ \begin{gathered} \left( {\vec{v}_{//} } \right)_{i} = \left( {\vec{v}_{i} \cdot \overrightarrow {{{\text{ISA}}_{\text{o}} }} } \right)\;\overrightarrow {{{\text{ISA}}_{\text{o}} }} \hfill \\ \left( {\vec{v}_{ \bot } } \right)_{i} = \vec{v}_{i} - \left( {\vec{v}_{i} \cdot \overrightarrow {{{\text{ISA}}_{\text{o}} }} } \right)\;\overrightarrow {{{\text{ISA}}_{\text{o}} }} \hfill \\ \end{gathered} $$

(5)

where i = [1, 2, 3], for each of the three markers. Magnitude of the parallel component of marker velocity, $\left| {\vec{v}_{{//_{i} }} } \right|$, corresponds to RB translation in ISA direction. In the present case study, this corresponds to velocity $\vec{v}_{//}$ from the screw motion.

The second step involves determining ISA location within the global reference frame. For each marker, a plane can be defined by its perpendicular velocity, acting as the normal vector to the plane, and its position, acting as a point on the plane [47]. ISA position is given by the line of intersection of the three planes as shown in Fig. 11.

Fig. 11

Top projection of planes formed by points M1, M2 and M3 with normal vectors $\vec{v}_{{ \bot_{1} }}$, $\vec{v}_{{ \bot_{2} }}$ and $\vec{v}_{{ \bot_{3} }}$, respectively. Position of ISA, ISAp, coincides with intersection of three planes

ISA position can be found using the intersection of only two of the three planes. For the case of analytical RB motion, the choice is arbitrary and, for this example, is chosen as the intersection of the planes defined by markers M1 and M2 and their respective perpendicular velocities. The equation of these two planes is given by the dot product of the normal vector to the plane and a point belonging to that plane:

$$ \begin{gathered} C_{i} = \left( {v_{ \bot } } \right)_{i} \cdot p_{i} \hfill \\ C_{i} = \left( {\left( {v_{ \bot x} } \right)_{i} \vec{i} + \left( {v_{ \bot y} } \right)_{i} \vec{j} + \left( {v_{ \bot z} } \right)_{i} \vec{k}} \right) \cdot \left( {x_{i} \vec{i} + y_{i} \vec{j} + z_{i} \vec{k}} \right) \hfill \\ \end{gathered} $$

(6)

where i = [1, 2] for markers M1 and M2, and C _i corresponds to the plane constant [47]. The intersection of these two planes is a line corresponding to ISA, which is represented by a single point, along the intersection line, and $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$. A single point belonging to the intersection line is determined using the followings linear system of equations:

$$ \left[ {\begin{array}{*{20}c} {\left( {v_{ \bot x} } \right)_{1} } & {\left( {v_{ \bot y} } \right)_{1} } & {\left( {v_{ \bot z} } \right)_{1} } \\ {\left( {v_{ \bot x} } \right)_{2} } & {\left( {v_{ \bot y} } \right)_{2} } & {\left( {v_{ \bot z} } \right)_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {C_{1} } \\ {C_{2} } \\ \end{array} } \right] $$

(7)

where (x, y, z) are the coordinates of a point belonging to the intersection line. This linear system consists of two equations and three unknowns. To solve for (x, y, z), the point coinciding with the intersection line and the xy-plane is determined, labelled (x _xy , y _xy , z _xy ). This sets the z-coordinate, z _xy , to zero, and the following system is linear equations is given:

$$ \begin{gathered} \left[ {\begin{array}{*{20}c} {\left( {v_{ \bot x} } \right)_{1} } & {\left( {v_{ \bot y} } \right)_{1} } & {\left( {v_{ \bot z} } \right)_{1} } \\ {\left( {v_{ \bot x} } \right)_{2} } & {\left( {v_{ \bot y} } \right)_{2} } & {\left( {v_{ \bot z} } \right)_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x_{xy} } \\ {y_{xy} } \\ 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {C_{1} } \\ {C_{2} } \\ \end{array} } \right] \hfill \\ \left[ {\begin{array}{*{20}c} {\left( {v_{ \bot x} } \right)_{1} } & {\left( {v_{ \bot y} } \right)_{1} } \\ {\left( {v_{ \bot x} } \right)_{2} } & {\left( {v_{ \bot y} } \right)_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x_{xy} } \\ {y_{xy} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {C_{1} } \\ {C_{2} } \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(8)

resulting in a linear system of two equations and two unknowns (x _xy , y _xy ) solved as from the inverse of the equation:

$$ \left[ {\begin{array}{*{20}c} {x_{xy} } \\ {y_{xy} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {v_{ \bot x} } \right)_{1} } & {\left( {v_{ \bot y} } \right)_{1} } \\ {\left( {v_{ \bot x} } \right)_{2} } & {\left( {v_{ \bot y} } \right)_{2} } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {C_{1} } \\ {C_{2} } \\ \end{array} } \right] $$

(9)

In the present case, (x _xy , y _xy , z _xy ) correspond to the origin of the global reference frame, (0, 0, 0), as $\vec{v}_{{ \bot_{1} }}$ of all markers is perpendicular to the radius from the applied angular velocity, at the centroidal axis, to each marker coordinate (Fig. 11). Therefore, the single point defining the position of the ISA, ISA_p, becomes

$$ {\text{ISA}}_{\text{p}} = (x_{xy} ,y_{xy} ,0) $$

Therefore, the coordinates of any point along the ISA is given by the multiplication of the unit vector $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ by a given constant, C, to the coordinates of the point ISA_p, as follows:

$$ \overrightarrow {\text{ISA}} = C\;\overrightarrow {{{\text{ISA}}_{\text{o}} }} + \overrightarrow {{{\text{ISA}}_{\text{p}} }} $$

Here, ISA coincides with the cylinder centroidal axis, as $\overrightarrow {{{\text{ISA}}_{\text{o}} }}$ direction is parallel to the z-axis and ISA_p coincides with the origin of the global reference frame, as shown in Fig. 12.

Fig. 12

Analytical cylinder; a applied screw motion; b ISA position and orientation

In the case where RB motion is unknown and marker position and velocity are measured, as is the case with stereophotogrammetric data, ISA position and orientation are computed as described above. The magnitude of segment parallel velocity, $||\vec{v}_{//} ||$, is determined from the projection of marker velocity $\overrightarrow {v}$ in the ISA direction. The magnitude of segment angular velocity, $||\vec{\omega }||$, can be computed from the magnitude of a given marker’s velocity and its radius to the ISA, as described in Eq. (2). The radius, $\overrightarrow {{r_{i} }}$, is defined as the vector spanning from the ISA to the position of the given marker while remaining perpendicular to the ISA. As the orientation of segment angular velocity is known, parallel to the ISA, the magnitude can be determined from the magnitude perpendicular marker velocity, $||\vec{v}_{ \bot } ||$, and the magnitude of vector $\overrightarrow {{r_{i} }}$. Therefore, segment motion about the ISA can be computed from marker displacement and ISA position and orientation.

Appendix 2: Euler–Cardan results

2.1 Euler angle: velocity ratios

Velocity ratios were computed using the Euler angles method, for comparison purposes with the ISA velocity ratios, and are shown in Tables 10, 11 and 12. Velocity ratios were computed as the ratio of segment translational velocity, computed from the origin of the segment local coordinate frame, to marker velocity resulting from segment angular velocity, computed from the segment Euler angles. Results shown are the mean velocity ratio, Mean, averaged for the duration of the downswing, the maximum value of the ratio, Max, and the velocity ratio at the instant of segment ω _max, expressed in percentage. For each subject, mean and standard deviation results were computed from the average over the five trials at each time increment. For the Mean ratio, standard deviation is the average value over all time increments, while the standard deviation for Max and ω _max ratios is given for the corresponding time increment.

Table 10 Subjects 1–5––Velocity ratio of the pelvis (mean ± standard deviation) computed as the ratio between the translational velocity of the origin of the pelvis local coordinate frame and the marker velocity from segment angular velocity, from Euler Angle method, expressed in percentage

Table 11 Subjects 1–5––Velocity ratio of the shoulders (mean ± standard deviation) computed as the ratio between the translational velocity of the origin of the shoulders local coordinate frame and the marker velocity from segment angular velocity, from Euler angle method, expressed in percentage

Table 12 Subjects 1–5––Velocity ratio of the left arm (mean ± standard deviation) computed as the ratio between the translational velocity of the origin of the left arm local coordinate frame and the marker velocity from segment angular velocity, from Euler Angle method, expressed in percentage

2.2 Euler angles: segment maximum angular velocity

Segment angular velocity was computed from segment Euler angles. Briefly, Euler angles were differentiated with respect to time to yield the x, y and z components of segment angular velocity. Maximum segment angular velocity and corresponding percent downswing are shown in Table 13.

Table 13 Subjects 1–5––maximum segment angular velocity, from Euler Angle method, and corresponding percent downswing (% DS)

Appendix 3: ISA analysis––segment angular velocity

Segment angular velocity was computed from marker velocity perpendicular to segment ISA, i.e. from rotation about segment ISA. Maximum segment angular velocity, with corresponding percent downswing, is shown in Table 14.

Table 14 Subjects 1–5––maximum segment angular velocity, about segment ISA, and corresponding percent downswing (% DS)