On the biomechanical analysis of the calories expended in a straight boxing jab

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].

Figure 1.

The motivation for the system to be modelled.

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 θ₄ = θ_c is controlled. The position vector loop around the linkage is given by

1.1

Differentiating, a velocity vector loop is generated

1.2

In component form, for a planar mechanism, we have, for the x-components of position

1.3

and for the y-components of position

1.4

where (for i = 1, 2, 3, 4) and all angles are measured anticlockwise from horizontal right (figure 1). The two unknowns are r₂ and θ₃. 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:

1.5

and for the y-components of velocity:

1.6

Figure 2.

Top view of an in-plane jab.

Figure 3.

Linkage diagram for an (horizontal) in-plane left jab.

2. Solution algorithm

A solution can be determined in closed form by writing

2.1

and taking the inner product r₃ · r₃,

2.2

which yields

2.3

Since r₁ · r₂ = 0 and using standard trigonometric identities we obtain

2.4

where θ₂ = 0 and θ₁ = 3π/2. Rearranging terms yields a quadratic equation

2.5

where a = 1 and

2.6

and

2.7

The quadratic equation can be solved for the length of r₂

2.8

This then leads to the solution of the angle θ₃

2.9

where, as stated before, θ₄ is controlled. The larger of the two roots in both equations is the correct one. Since , , , , , , we have

2.10

and

2.11

In matrix form

2.12

which yields

2.13

3. Energy principles

One can immediately post-process the kinetic energy in the system as

3.1

where in this specific case

3.2

and

3.3

where 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

3.4

where 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 + Δt, configuration 2) to yield

3.5

where 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 r₂ reaches a critical length to make contact. We can then immediately represent the incremental work as

3.6

A relatively easy parametrization of the losses is (when in contact and the jab is moving forward)

3.7

where 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 integrate¹

3.8

This 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: r₁ = 0.05 m,

• — link 2 (location to fist) length: r₂ = 0.25 m (starting),

• — link 3 (forearm) length: r₃ = 0.3 m,

• — link 4 (upper arm) length: r₄ = 0.25 m,

• — mass of link 2 (fist): m₂ = 0.65 kg,

• — mass of link 3 (forearm): m₃ = 1.5 kg,

• — mass of link 4 (upper arm): m₄ = 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, 10², 10³, 10⁴, 10⁵, 10⁶. As the results in table 1, values of C ≤ 10³ burn extremely low numbers of calories. Between 10³ ≤ C ≤ 10⁵ is a realistic range of parameters, while C > 10⁵ 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 10³ ≤ C ≤ 10⁶. 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.

Figure 4.

(a–f) Sequence of a punch to a bag at equal one-sixth interval times: t = 0, T/6, T/3, T/2, T/6, T, where T is the total time. The fist crossing into the box indicates that contact with the bag is taking place, accruing dissipation. The box is not the bag, it merely indicates when dissipation is taking place. (Online version in colour.)

Table 1.

The calories expended for various values of the loss-coefficient, C = 0, 1, 10, 10², 10³, 10⁴, 10⁵, 10⁶.

loss coeff.: C	kilocalories/jab	total kilocalories (720 jabs)
0	0.0134	9.711
1	0.0135	9.723
10	0.0136	9.839
102	0.0165	11.934
103	0.0480	34.591
104	0.3627	261.155
105	3.5094	2526.797
106	34.9767	25183.215

Figure 5.

The incremental work (in joules), for various values of the loss-coefficient, (a) C = 10, (b) 10², (c) 10³ and (d) 10⁴. The spike occurs when contact is made with the object, with the decrease resulting upon retraction of the ‘fist’.

Remark. —

A variety of popular articles indicate that approximately 500–1000 calories are burned in a 1-h workout with a heavy bag for a 75–100 kg male [1–6]. This somewhat qualitative data would be interpreted for entire body movement, which we discuss next.

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:

5.1

and y-components

5.2

These can be linearized for a Newton-iteration (i), for the x-components

5.3

and for the y-components

5.4

In matrix form

5.5

with the solution being

5.6

This 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.

Figure 6.

Hypothetical multiple coupled vector loops that must be solved numerically.

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.