## Abstract

Boxing and related sports activities have become a standard workout regime at many fitness studios worldwide. Oftentimes, people are interested in the calories expended during these workouts. This note focuses on determining the calories in a boxer's jab, using kinematic vector-loop relations and basic work–energy principles. Numerical simulations are undertaken to illustrate the basic model. Multi-limb extensions of the model are also discussed.

## 1. Introduction: kinematic vector loop representation of a jab

Fitness enthusiasts are keenly interested in the calories expended during non-standard work regimes involving martial arts, boxing and related sports [1–6]. As an example of how to determine the calories expended in a simple straight boxer's jab (figure 1) from basic principles, we employ a combined kinematic and energy analysis, by drawing on methods used in the robotics literature [7–16].

Accordingly, consider the idealization of a jab illustrated in figures 1–3 as a linkage. Viewing the boxer from the top, we can analyse the motion of the components of a linkage by applying a closed vector loop that traverses the upper arm, forearm and fist, generating a slider crank mechanism capable of describing a straight left jab. It is made up of a block mass (fist) with mass *m* attached by two rigid rods (upper arm and forearm). The angle *θ*_{4} = *θ*_{c} is controlled. The position vector loop around the linkage is given by
1.1Differentiating, a velocity vector loop is generated
1.2In component form, for a planar mechanism, we have, for the *x*-components of position
1.3and for the *y*-components of position
1.4where (for *i* = 1, 2, 3, 4) and all angles are measured anticlockwise from horizontal right (figure 1). The two unknowns are *r*_{2} and *θ*_{3}. The velocity unknowns are and , and can be subsequently found from differentiating the component equations of equation (1.1), yielding, for the *x*-components of velocity:
1.5and for the *y*-components of velocity:
1.6

## 2. Solution algorithm

A solution can be determined in closed form by writing
2.1and taking the inner product **r**_{3} · **r**_{3},
2.2which yields
2.3Since **r**_{1} · **r**_{2} = 0 and using standard trigonometric identities we obtain
2.4where *θ*_{2} = 0 and *θ*_{1} = 3*π*/2. Rearranging terms yields a quadratic equation
2.5where *a* = 1 and
2.6and
2.7The quadratic equation can be solved for the length of **r**_{2}
2.8This then leads to the solution of the angle *θ*_{3}
2.9where, as stated before, *θ*_{4} is controlled. The larger of the two roots in both equations is the correct one. Since , , , , , , we have
2.10and
2.11In matrix form
2.12which yields
2.13

## 3. Energy principles

One can immediately post-process the kinetic energy in the system as
3.1where in this specific case
3.2and
3.3where all velocities are known from the computations of the previous section. In a general case, we would need to include the potential energy due to gravity
3.4where *g* is gravity and *h*_{i} is a given datum height (from a reference). *However, this is not needed for a horizontal in-plane jab, although we will keep the term for completeness of the formulation*. From standard work–energy principles, we equate the sum of the kinetic and potential energy in an arbitrary ‘configurational state 1’ (at time = *t*), plus the work done from state 1 to a later (incremental) state, ‘configurational state 2’ (time = *t* + Δ*t*, configuration 2) to yield
3.5where is the incremental work done in moving the boxer's arm and fist, and are the losses due to contact with the bag. This is enacted with *r*_{2} reaches a critical length to make contact. We can then immediately represent the incremental work as
3.6A relatively easy parametrization of the losses is (when in contact and the jab is moving forward)
3.7where *C* is a loss coefficient per unit length, and otherwise (when the fist not in contact with the bag or the jab is retracting). We can then integrate^{1}
3.8This gives the energy in joules. We can then convert into calories by *K*_{c} = *W*^{tot}/4.184 and into commonly used layman jargon ‘calories’, which are actually ‘kilocalories’ by kilo-*K*_{c} = *W*^{tot}/4184.

## 4. A numerical example: an hour-long workout

As an example, we consider the following parameters for an in-plane jab by using the average values of the range of a 75–100 kg male, from de Leva [17], Tozeren [18] and Plagenhoef *et al.* [19]:

— angular motion:

— link 1 (offset) length:

*r*_{1}= 0.05 m,— link 2 (location to fist) length:

*r*_{2}= 0.25 m (starting),— link 3 (forearm) length:

*r*_{3}= 0.3 m,— link 4 (upper arm) length:

*r*_{4}= 0.25 m,— mass of link 2 (fist):

*m*_{2}= 0.65 kg,— mass of link 3 (forearm):

*m*_{3}= 1.5 kg,— mass of link 4 (upper arm):

*m*_{4}= 2.5 kg,— location of the target (bag),

*r*_{x}= 0.4 m.

The angular acceleration of link 4 during extension (from rest) is given by , and when retracting by . The total simulation time for a quick jab was approximately *T* = 0.3 s. The author is an amateur boxer, and these times and rates were determined from his own measurements. The linkage sequence is shown in figure 4. The loss coefficient (measured in joules/metre) is the one parameter that was varied, as it would depend on the type of obstacle being hit. We varied from ‘shadow punches’ (no resistance) to extremely heavy resistance: *C* = 0, 1, 10, 10^{2}, 10^{3}, 10^{4}, 10^{5}, 10^{6}. As the results in table 1, values of *C* ≤ 10^{3} burn extremely low numbers of calories. Between 10^{3} ≤ *C* ≤ 10^{5} is a realistic range of parameters, while *C* > 10^{5} results in expenditures that are unrealistic. The total kilocalories used in a single jab was computed, then to illustrate a ‘mock’ workout, we estimated a workout associated one jab every 5 s, thus 12 per min, thus 720 in 1 h. The incremental work done by the system is shown in figure 5 for the range 10^{3} ≤ *C* ≤ 10^{6}. One can see that upon contact with the body, the work rate jumps dramatically. It is important to emphasize that this is for a simple straight jab, and does not take into account other body motion. In order to more accurately compute biological energy expenditure one can use the concept of internal and external energy using muscle models, for example, found in [20,21]. The use of this type of framework to analyse more complex sports kinematics and the associated calorie expenditures is currently under investigation by the author. However, this will also require extensions to multi-limb movement, which we next discuss in the summary and extension section.

## 5. Summary and extensions for complex systems

The framework developed provides an analysis of a simple jab. Of course, in boxing and other contact sports, the motions can be much more complex, and will not lend themselves to closed form solutions. In particular, if one were to compute the dynamics of an entire body (figure 6), coupled vector loop systems will occur, and one must resort to numerical approaches. For example, in order to illustrate how numerical procedures are applied to such systems, consider again equation (1.1), which can be expressed in terms of its *x*- and *y*-components to form a nonlinear set of equations, which can be written as:
5.1and *y*-components
5.2These can be linearized for a Newton-iteration (*i*), for the *x*-components
5.3and for the *y*-components
5.4In matrix form
5.5with the solution being
5.6This can be solved at every increment, and is widely used in the analysis of robotic and kinematic systems involving large systems of coupled vector loops, such as those found in [7–16]. This multi-limb approach, coupled to the concept of internal and external energy using muscle models [20,21], is under current investigation by the author.

Finally, a further point of investigation is the correlation of body blows to accelerated atherosclerosis and general artery stenosis in ageing boxers (as well as in participants in other contact sports). The approach is to couple recent models found in [22–25], which describe the accumulation of microscale materials at solid–fluid interfaces in biological channels. This is often the initial stage of biochannel occlusive growth processes [26–30].^{2} In those models, the approach is to construct rate equations for the accumulation of suspended materials in the bloodstream at the solid–fluid interface. The accumulation of material subsequently reduces the cross-sectional area of the channel, which can lead to dementia-like symptoms commonly associated with older boxers (such observations date back, at least, to Mawdsley & Ferguson [45] and with relatively recent data collected in Krishnan [46]). The generation and deposition of calcium and fatty deposits on channel walls is an open question of increasing interest [47–50], and the repeated blows from contacts sports has been hypothesized to enhance such growth. A future objective is to use kinematic models, such as the one developed here, to simulate a wide range of induced forces involving fist-to-head and fist-to-chest force calculations and to determine the connections to possible accelerated biological channel growth.

## Competing interests

We declare we have no competing interests.

## Funding

We received no funding for this study.

## Footnotes

- Received March 1, 2017.
- Accepted March 20, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.