Error in jump height estimation using the flight time method: simulation of the effect of ankle position between takeoff and landing

Biomechanical model

A four-segment model (torso, thigh, shank, and feet) (Winter, 2009) was created to evaluate the vertical CoM distance from the floor (Fig. 1). Additionally, the tiptoe-to-ankle (lateral malleolus) horizontal distance was set at 78.7% of the foot length (Tilley & Henry Dreyfuss Associates, 2001) (Fig. 2). The motion of the model was constrained to the sagittal plane while it stood on a single leg with doubled expected mass. The kinematic chain origin was set at the tiptoes, and a set of 2D rotational matrices (Robertson et al., 2013) was used to create the expected motion of the ankle, knee, and hip joints. The kinematic chain rotations provided the CoM position for each body segment (foot, shank, hip, and torso). The whole-body CoM was calculated from a weighted sum of the segment’s CoM position and mass, with CoM vertical displacement exclusively resulting from changes in ankle position. Therefore, the maximum change in body posture between jump takeoff and landing was set at landing with feet flat (angle joint at 0°).

Simulated jumps and research variables

We used a continuum of 100 body heights ranging from 1.435 m (3 standard deviations (SD) below the women’s average) to 1.984 m (3 SD above the men’s average) (WHO, 2007) , which was required to estimate foot length (Winter, 2009) (15.2% of the stature) and subsequently ankle position. The simulations also accounted for jump heights of 0.10, 0.20, 0.30, and 0.40 m allowing the simulation of 16,400 jumps (100 stature × 41 angle positions × 4 jump heights). The takeoff velocity and ascending time were calculated using the constant acceleration equations. The error in jump height estimation was computed for each condition. The difference in CoM position due to a change in ankle position was defined as any difference between landing and takeoff (nomenclature definitions at Table 1): (1)${\displaystyle h_{diff}=h_l-h_{to}.}$

The ascending time (t_a) is directly related to jump height (h), the takeoff velocity (V_to), and the true flight time (FT_true), i.e., the CoM height is the same at the takeoff and landing of the jump. Therefore,

(2)${\displaystyle t_a=\sqrt{\frac{2.h}{g}},}$ (3)${\displaystyle F{T}_{true}=2.t_a,}$ (4)${\displaystyle V_{to}=g.t_a.}$

Consequently, the measured FT when there was any change in ankle position was estimated as: (5)${\displaystyle FT=\frac{V_{to}+\sqrt{V_{to}^2-2.g.h_{diff}}}{g}.}$

The percentage difference in jump height estimations was defined as h_error, where $\hat{\mathrm{h}}$ is the approximation using the measured FT.

(6)${\displaystyle \hat{\mathrm{h}}=\frac{F{T}^2.g}{8},}$ (7)${\displaystyle h_{error}=\left(\frac{\hat{h}-h}{h}\right).100.}$

When there is any change in ankle position (or any change in body posture) between takeoff and landing, the presented equations can be manipulated to obtain FT_true from FT and t_diff (time difference due to change in body posture), as follows:

(8)${\displaystyle F{T}_{true}=FT-t_{diff},}$ (9)${\displaystyle t_a=\frac{1}{g}.\left(\frac{g.FT}{2}-\frac{h_{diff}}{FT}\right),}$ (10)${\displaystyle h=\frac{1}{2.g}.{\left(\frac{g.FT}{2}-\frac{h_{diff}}{FT}\right)}^2.}$

Obtaining h_diff is challenging without a motion capture system. Therefore, we evaluated the potential of estimating h_diff (${\hat{h}}_{diff}$) based on ankle-toe length (l_at) estimated from stature (H), and ankle position: ankle angle at takeoff (α_to), and landing (α_land), acquired from 2D video analysis (Fig. 2).

(11)${\displaystyle l_{at}=\sqrt{{\left(0.039H\right)}^2+{\left(0.152\times 0.787H\right)}^2}=0.126H,}$ (12)${\displaystyle {\hat{h}}_{diff}=l_{at}\left(sin\left({\alpha }_{to}+\beta \right)-sin\left({\alpha }_{land}+\beta \right)\right),}$

therefore, the “corrected” jump height (${\hat{h}}_c$) can be estimated as: (13)${\displaystyle {\hat{h}}_c=\frac{1}{2.g}.{\left(\frac{g.FT}{2}-\frac{{\hat{h}}_{diff}}{FT}\right)}^2.}$

Auxiliary tool

Using this calculator https://www.lptf.unb.br/calculadora the estimated jump can be automatically corrected by inserting (1) a change in ankle position and (2) stature or foot length or, ideally, ankle-toe distance. For changes in ankle position at landing, we have provided information for setting up the recording device (e.g., smartphone) and video processing using the freely available Kinovea software (https://www.kinovea.org/). The dataset used for this research is also available for consulting.