A Simple Model of the Handstand

Warning: This blog post contains math. If you don’t like math and tend to break out in hives at the sight of variables and equations, you are more than welcome to skip to the figures in the results.

Disclaimer: The model described below is just that, a model. With this comes assumptions and approximations that are a necessary part of the process. Most of these are stated explicitly, but including an exhaustive list would be unreasonable. I encourage you to attempt to understand the general trends and phenomena, while accepting that modeling requires some hand waving and approximation.

Introduction

I'll occasionally hear that it is more difficult to do a handstand when you're tall. Upon hearing this, my practical handbalancing mind goes one direction, and my engineering mind goes in another direction, which is to explore this phenomenon with whatever tools I have. I realized that I could explore this from a more analytic perspective using a very simplified model of the handstand. This post will detail this model and will explore how body mass, height, and hand length affect the necessary force that is required to balance a handstand. The model and approach bring to light some important assumptions and concepts, and these are discussed in regards to their practical significance and practical application when doing a handstand.

Methodology

This section details the biomechanical model, which is a simple inverted pendulum model.. While the model of the body is rather simplified, there is a bit more detail given to the hand/fingers, as this entity is significant to managing balance.

Inverted pendulum model

For this static analysis, we will use a basic inverted pendulum model. In this model, as in a handstand, the individual has their hands flat on the floor, the arms, body, and legs are inverted and straight, and the point of rotation of the pendulum is the wrists. The body and limbs are simplified to straight segments and the joints are simplified to single points of rotation. We will use the following assumptions:

• Assumption 1: The body remains rigid and the only joint that can rotate is the wrist joint. Balancing in a handstand does require whole-body strength to maintain a straight body, but this is a simplified model, and we will assume the individual has this strength. In practice, this must be trained.

• Assumption 2: In light of Assumption 1, balance happens through the application of positive torque through the wrists via an application of purely downward-directed force at the fingertips and through a rigid hand/finger complex. There are no other body movements that assist with balancing.

• Assumption 3: Considering the two above assumptions, we will look only at controlling overbalance, where the body leans towards the fingertips. This is possible to control through only the use of the hand/finger complex with nothing else moving in the body.

In addition to these assumptions, it is important to acknowledge that biomechanical modeling requires simplifications and acceptance that there are real-world details that are left out, not out of ignorance or carelessness, but out of the fact that modeling such details may not be worth the effort. With that, I will take some modeling liberties that are rather standard. Additionally, numbers will be rounded in some cases and this rounding likely has minimal impact on the results considering that individual variability and the simplification of the model likely have a greater impact on the accuracy of the model.

'Balancing' a handstand is a dynamic process, however, to simplify things, we will only consider a static analysis. This means that we are interested in understanding the necessary control force of the hands/fingers, which generates a torques at the wrists, that is equal and opposite to the torque generated by the inverted body as it tilts to overbalance. In other words, we will identify the control force at the fingertips that is necessary to statically balance a handstand at specific angles and without the intention to return the handstand towards vertical. Placing the handstand back towards vertical from this static tilt would simply require greater force through the fingertips to generate greater torque through the wrists.

Figure 1 shows the free-body diagram of the inverted pendulum model. The center of mass is located at a distance b from the wrist joint. Due to the title angle alpha (α) and gravity, the body mass M falls, which generates a moment about the wrists. To statically balance the handstand at some angle α, a force Ftotal is applied at the fingertips (with hand length lhand), which generates a moment at the wrists that is equal and opposite to that generated by the falling center of mass of the individual. Note that αmax is the maximal angle at which the body can tilt (i.e., limit of stability), which occurs when the horizontal movement of the center of mass due to a tilt is equal to the hand length.

Figure 1. Inverted pendulum model of the handstand.

Mathematically, the equal and opposite moments of the inverted body and of the hand/fingers can be described by:

Equation 1: M * g b sin(α) = τ total = F total l hand

where M * is the body mass less the mass of the hands, g is gravity, b is the location of the aggregate body center of mass, and α is the tilt angle. τ total is the total wrist torque, which equivalent to F total , the downward force applied at the fingertips from both hands, at a lever arm that is the length of the hand (l hand ). (I apologize about the equation format; Squarespace doesn't have a good equation editor, such as something akin to LaTeX.)

Anthropometry

Body proportions are a critical component to appropriately placing the center of mass of the body. The source for anthrompometric data will be those that are most popularly cited from Winter (Winter, 2009), but that are originally from Drillis and Contini (Drillis and Contini, 1966). (I acknowledge that the body mass distribution parameters that Drillis and Contini report may have changed substantially since 1966, but that's a discussion for another day.)

The body is represented as five groups of segments: hand/finger complex, arms (including upper arm and forearm), torso (including head and neck for body mass parameters), leg (including upper and lower leg), and foot. Of course, there are two each of the hand/finger complexes, the arms, the legs, and the feet. The side view of the model is shown in Figure 2 along with the segment lengths as proportions of the total body height H. Table 1 also shows the segment length proportions in addition to the segment masses as proportions of total body mass M and the center of mass locations of each segment relative to the joint closest to the floor when the body is oriented as in Figure 2. Based on these data, the center of mass of the entire body (not included hands) in this specific position can be computed, resulting in an aggregate center of mass that is located at approximately 55.5% of one's height from the floor (Figure 2).

Figure 2. Segment lengths of the body relative to body height H. The aggregate center of mass of the body is located at 0.555H.

Table 1. Body segment lengths as a proportion of height H, body segment masses as a proportion of total body mass M, and center of mass locations of each segment relative to the floor when the body is inverted as in Figure 2. GHJ = glenohumeral joint, GT = greater trochanter. *See discussion on hand length below. Note that this hand length is not used in the model, but it is reported here.

Equation (1) states that the amount of torque necessary to sustain some degree of tilt is dependent on the hand length. Hand length in this content is critical and it is more involved to determine. To illustrate this, I will provide three definitions of hand length:

• Actual hand length: This is the length of the hand as measured from the base of the hand (crease of the wrist) to the end of the middle finger.

• DC hand length: This length is reported by Drillis and Contini (1966) as a proportion of height. However, this length is measured from the ulnar styloid to the end of the middle finger, and thus it is greater than actual hand length as defined above. This value is longer than what is actually used to balance in a handstand and thus it cannot be used directly. This value will be used to determine an appropriate scaling factor for an average person based on my measurements and proportions.

• Effective hand length: This is the hand length that is an average value of the effective lever arm of the hand that is used to balance a handstand. More details on this length will be provided below, though it should be noted that this value is smaller than the actual hand length due to averaging the individual lever arms of all digits.

Considering that the hand length is explicitly represented in Equation (1), this will be a quantity of interest in the analysis. And, we will be a bit more precise in how we consider this quantity.

As stated in Assumption 2, balancing a handstand is a result of producing a torque at the wrists, where this torque is generated by applying a downward-directed force through the fingertips of both hands (via a rigid hand/finger complex). The lever arm of this rigid hand/finger complex isn't quite the actual hand length as defined above. Rather, each digit has a different length as measured from the base of the hand. Furthermore, each digit contributes to some portion of the total downward-directed force for balancing, and the force distribution across the fingers is likely not even.

This is where the effective hand length comes into play, which will be computed using a force contribution-weighted average of digit lengths, where the distribution of forces at the fingertips is taken from pilot data I obtained a few years ago from a plantar pressure platform (click here to see this). The digit lengths (from the base of the hand to the point of force application into the floor for each digit, as measured in line with the direction of balance/sway) and the proportion of force contribution for each digit are shown in Table 2.

Table 2. Digit lengths and digit force distribution during a handstand.

Averaging the digit lengths based on their force contribution results in an effective hand length of 14.88 cm for my hand and hand/finger usage. My actual hand length is 19 cm, and the effective hand length is therefore about 78% of my actual hand length. We will assume that this proportion of my effective hand length to actual hand length is consistent across individuals for the sake of this model, but I acknowledge that individuals have different digit length, different distribution of forces at the digits, and different digit configurations (i.e., spread of the digits or tenting perhaps), and this would have an effect on the effective hand length and the results of the model. The calculations using the Equation (1) will use the effective hand length, though all analyses will be reported based on an actual hand length (which you can measure on yourself) as individuals would likely rather think about the results relative to their actual hand length.

Analysis

Before getting to a numerical analysis, we can learn some things simply by looking at Equation (1). The primary direction of this analysis is in regards to the amount of force application that is necessary to sustain some degree of tilt given anthropometric measures of mass, height, and hand length. In light of this, and by rearranging terms in Equation (1), we have:

Equation 2: (1 / l hand ) M * g b sin(α) = F total

This form tells us that, for some amount of tilt α, the amount of force that the hand/finger complexes must apply increases with mass and height, and decreases with hand length. (As much as some people in the handstand world would like to think, gravity is not a variable.) In other words, when either or both mass and height increase, and for a set hand length, more force is required. Additionally, for a set mass and height, as hand length increases, the amount of force required decreases. Alternatively, when hand length decreases, say in the case of tenting the fingers, the amount of force required increases.

This basic understanding of how things change is useful, but numbers can say a lot as well. Univariate analyses of each of the four variables in Equation (2) is possible, but the interpretation would be rather muddy. Another way to look at things, using the notion that each individual has a set mass, height, and hand length, is to use metrics that rely on these quantities. We will define two metrics, MH and MH/l:

MH = mass x height

MH/l = mass x height / l hand

where mass is in kilograms (kg), and where height and l hand are in meters (m). (To convert, 0.4529 kg = 1 lb and 0.0254 m = 1 inch.) There are units attached to these metrics (kg x m for MH, and kg for MH/l), but let’s leave them off to avoid confusion. As an example, I am 140 lbs (63.4 kg), 64” (1.63 m), and my actual hand length is 0.19 m. This results in metrics of MH = 63.4 x 1.63 = 103.3 and MH/l = 63.4 x 1.63 / 0.19 = 543.9. (To reiterate a note above, although MH/l uses the actual hand length and is used for the presentation of results, the effective hand length is used when computing necessary hand/finger forces.)

Thus, each individual will have a specific value of MH and MH/l. However, if you tent you fingers, the value of lhand will be your tented hand length. I don’t tent my fingers, but if I did, my hand length would come to about 0.165 m, which would result in a tented MH/l metric of 626.3. Keep reading and you’ll see why these numbers are useful.

The numerical analyses that we will look at are as follows:

1. Hand/finger complex force as a function of handstand tilt angle α for values of MH/l ranging from 400 to 1200.
2. Hand/finger complex force as a function of hand length for a set tilt angle of 3 degrees for values of MH ranging from 60 to 200.
3. Maximal handstand tilt angle that is possible based on height and lhand.

The values I came up with for MH/l and MH were roughly based on large and small values of mass, height, and hand length for individuals that would likely be doing handstands. Roughly, these values are mass ranging from 36 to 113 kg, height ranging from 1.37 to 1.93 m, and actual hand lengths ranging from 0.07 to 0.25 m. It’s important to state that these values used were estimated off the top of my head, and they are not representative of a distribution of values from individuals in the handstand world. I’ve thought about putting out an informal poll to obtain data from handstanders to understand where we all fall in these metrics…that could happen.

Results

Figures 3 and 4 use a certain type of presentation, and it’s worthwhile explaining this in full. The black lines represent increments of 100 for the MH/l metric. You, as an individual, have a unique mass M, height H, and hand length l hand , and thus have a unique MH/l metric and a unique MH metric, which lie on a black line or between two black lines. Think of these like isobars (same pressure) on a weather map. Additionally, the y-axis represents the total force of both hands that must be applied at the fingertips and through the rigid hand/finger complex.

Control force as a function of MH/l

Figure 3 shows the how total hand control force varies with handstand tilt angle and by MH/l. For any value of MH/l, the total hand force seems to linearly increase with handstand tilt angle; the trend is actually sub-linear, though it just seems linear in this region with relatively small angles. Speaking more to the body parameters, individuals that have a high mass, that are tall, and that have short hands have large values of MH/l and need relatively more hand force compared to those with low mass, that are short, and that have large hands (and thus, have small values of MH/l). My flat fingered hand across all tilt angles is the dashed red line, with an MH/l value of 543.9 (thus it lies between the 500 and 600 black lines). My tented hand would be the dashed blue line, which requires 15.4% more force for any angle of tilt for a hand length reduction of 13.2% due to tenting the fingers. It should be noted that all of the more extreme tilt angles plotted are not universally achieveable as MH/l is varied, but most of the smaller angles are. The limit of stability is discussed later in the results.

Alternatively, we can set the amount of hand force necessary to balance, say at 500 N. When we look at the extremes on the graph, an individual with an MH/l of 400 could potentially tilt to 10 degrees (if they are within the limit of stability). An individual with an MH/l of 1200 would only be able to sway to just beyond 3 degrees.

It is important to note that individuals with very different bodies can have the same MH/l metric. For example, someone that is rather tall, thin, and has a rather large hand (M = 77 kg, H = 1.83 m, l hand = 0.22 m) has an MH/l of about 640, which is very similar to someone that is short, proportionally thin, with a smaller hand (M = 46 kg, H = 1.52 m, l hand = 0.11 m). For a total hand force of 500 N, both of these individuals could sway to about 6 degrees.

Figure 3. Total hand control force as a function of handstand tilt angle and MH/l. The red line shows where I fall on this plot where MH/l = 542.5 (body mass M = 63.4 kg, height H = 1.63 m, hand length l = 0.19 m). The dashed blue line shows where I fall if I tent my fingers, which reduces l to 0.165 m (MH/l = 626.2).

Control force as a function of MH

Figure 4 shows how if we set a tilt angle, in this case to 3 degrees, the hand force required to balance this tilt changes in a non-linear manner as a function of hand length. As with Figure 3, the black lines represent curves for set values of MH, and together with a specific hand length, each individual is a single point on the plot. We see that as hand length gets very small, the required total hand force increases more than for larger hand lengths. As with Figure 3, the fingers could be extended and flat or they could be tented, and these two cases are shown in the figure to illustrate the the necessary increase in force when the finger position is modified.

Figure 4. Total hand control force as a function of hand length l and MH for a 3 degree tilt of the handstand. The red dot shows where I fall on this plot where MH = 103.1 (body mass M = 63.4 kg, height H = 1.63 m) and hand length l = 0.19 m. The blue dot shows where I fall with a tented hand length of 0.165 m.

Limit of stability

Figure 5 shows the maximum angle of tilt for any ratio of hand length to height. The maximum tilt angle here is based solely on hand length and height, and maximum tilt angle in practice is likely limited more by hand/finger strength as greater hand/finger strength is require for greater body masses. Furthermore, maximum tilt angle is also limited by the ability of the individual to maintain a straight/tight position when in overbalance. This figure assumes the individual has sufficient hand/finger strength, body tightness, and psychological comfort in overbalance.

Figure 5. Maximum tilt angle as a function of hand length to height ratio.

For my hand length of 0.19 m and height of 1.63 m, my hand length/height ratio is about 11.7, resulting in a maximum tilt angle of around 9.5 degrees. This is interesting because it can be used as a form of validation for this model. My maximum tilt angle in practice is approximately 10 degrees as roughly measured from a video. Given the similarity of these values, this should provide some credibility to even this very simplified model of the handstand. When I tent my fingers and my hand length is reduced to 0.165 m, my maximum tilt angle in Figure 5 decreases to just over 8 degrees.

To be honest, there's nothing really groundbreaking here. The effects of mass, height, and hand length are rather intuitive, as is the notion that as you tilt more, you have to work harder with your hands/fingers. However, I think this exercise is useful simply to see explicitly how things change and perhaps to see specifically where you fall on these curves relative to other individuals of different sizes. As mentioned, it would be interesting to understand the distribution of mass, height, and hand length for those that regularly train handstands with the primary intention of understanding if handbalancers generally have larger than proportional hands. Taking this a bit further, it would be interesting to understand how these quantities affect handstand stability and overall ability level from a quantitative perspective.

This model assumes that the hand/finger control force is directed downward and into the floor, regardless of whether the individual tents or not. Many individuals, and even those that are at an intermediate handstand level, are deficient in using their fingers in this manner. Instead, their fingers tend to grip and to retract as they balance, resulting in a force direction that is in part downward but that is also directed towards their palm, the second of which is not useful for balancing a handstand. This might be a topic for the future as proper finger usage is critical and does not get paid sufficient attention.

One related and important topic that this discussion did not get into was hand/finger strength in different finger positions and relative to body proportions. There is the question of whether the tented finger position can provide hand/finger downforce that more than compensates for the reduced lever arm relative to flat fingers. To be honest, I don’t believe there is much literature out there where we can appropriately draw conclusions for the handstand. The vast majority of research is aimed at general hand/finger usage and likely looks at flexed fingers used in a gripping fashion rather than extended (or tented) fingers used as they are in a handstand. In my mind, it would not be prudent to take studies that look at gripping and apply their findings to finger use in a handstand. I have run across a few studies that do study finger force with extended fingers and a rather open hand posture, and these works could have some information that is loosely relevant (Danion et al, 2000; Li et al, 2000; Shinohara et al, 2004).

Conclusion

This discussion was aimed at understanding or debunking the notion that being tall makes handbalancing more difficult. This simplified belief doesn’t consider all factors that have an effect on the required hand/finger force for balancing a handstand. In particular, hand length and body mass should be considered as well. However, in the vast majority of cases, none of these should be excuses for ability. If you spend sufficient time working the right things in the correct manner, you will improve. Hand/finger strength can be a focus and honestly should not just be left to come naturally when training handstands.

Am I suggesting that you don’t tent in order to allow for more sway or to reduce how much force you need to apply? I’ll leave that up to you. There are plenty of high level handbalancers that do tent their fingers, and plenty that don’t tent. My opinion is that being able to sway a comfortable amount is extremely useful for a couple of reasons. The first reason is that it allows one to have calm balance where the tilt of the handstand comes and goes and is corrected for in a very proactive and confident manner; you accept that sway will happen and are capable of managing it. It’s a peaceful balance where you have this comfort in variability. This is in contrast to an anxious balance where the individual is limited in their range of sway and where they must be very attentive in every single way. This is not comfortable in my opinion. The second reason is that being able to sway a good amount is useful in learning higher level skills as your body is challenged in various ways. Having the strength and familiarity to sustain, correct for, and work out of a bit of a tilt will allow you to spend more time in a zone of skill acquisition and motor learning, as opposed to falling at the slightest tilt. It is this time spent in a quality place will allow for the most efficient progress.

References

Danion F, Latash ML, Li ZM, and Zatsiorsky VM (2000). The effect of fatigue on multifinger co-ordination in force production tasks in humans. J Physiology, 523.2, pp.523-532.

Drillis R and Contini R (1966). Body Segment Parameters, Rep. 1163-03, Office of Vocational Rehabilitation, Department of Health, Education, and Welfare, New York.

Li ZM, Zatsiorsky VM, and Latash ML (2000). Contribution of the extrinsic and intrinsic hand muscles to the moments in finger joints. Clin Biomechanics, 15, pp. 203-211.

Shinohara M, Scholz JP, Zatsiorsky VM, and Latash ML (2004). Finger interaction during accurate multi-finger force production tasks in young and elderly persons. Exp Brain Res, 156: 282-292.

Winter DA (2009). Biomechanics and Motor Control of Human Movement. John Wiley & Sons, Inc. Hoboken, New Jersey, pp. 82-86.