biomech

What is energy?

The capacity to do work. In biomechanics it exists as potential energy (PE), translational kinetic energy (TKE), and rotational kinetic energy (RKE).

What is the formula for potential energy?

PE = mgh. Where m = mass (kg), g = 9.81 m/s², h = height above reference point (m). Units = joules (J).

What is the formula for translational kinetic energy?

TKE = ½mv². Where m = mass (kg) and v = velocity (m/s). Units = joules (J). Note: velocity must be squared.

What is the formula for rotational kinetic energy?

RKE = ½Iω². Where I = moment of inertia (kg·m²) and ω = angular velocity (rad/s). Units = joules (J).

What is Total Mechanical Energy (TME)?

TME = PE + TKE + RKE = mgh + ½mv² + ½Iω². The sum of all three forms of mechanical energy at any instant.

What is mechanical work?

Work = force × displacement in the direction of that force. W = F·d·cosθ. Units = joules (J). Work is the change in energy.

What is power?

Power is the rate of doing work. P = dE/dt = F·v·cosθ. Units = watts (W = J/s).

What is concentric muscle contraction in terms of work?

The muscle shortens and does positive work on the outside world. The muscle moment and joint motion are in the same direction. Work is positive (+W).

What is eccentric muscle contraction in terms of work?

The muscle is lengthened by an external force. Negative work is done on the muscle — it acts as a brake absorbing energy. Work is negative (−W).

What is isometric muscle contraction in terms of work?

No displacement occurs so zero mechanical work is done despite metabolic energy being consumed. W = F × 0 = 0. Energy is converted to heat.

What is the 1st Law of Thermodynamics and why does it matter?

Energy cannot be created or destroyed, only transformed. This means total mechanical energy is conserved — energy changes form but the total remains constant.

What is the 2nd Law of Thermodynamics and why does it matter for biomechanics?

When energy transforms, some is always wasted as heat (entropy). This means external mechanical measurements always underestimate true total muscle work — some energy always escapes as heat.

What is Fenn's method?

The fraction approach to mechanical energy. Tracks how TME is partitioned between PE, TKE, and RKE at each instant. Work done = change in TME between two time points.

What is the key insight of Fenn's method?

Energy directed into one form is unavailable for another. For example, a rolling object must split energy between TKE and RKE — so it moves slower than a sliding object of the same mass.

What is Elftmann's method?

A method for calculating joint power. Pj = Mj × (ω1 − ω2) = Mj × ωj. Where Mj = joint moment and ωj = relative angular velocity between the two adjacent segments.

When does Elftmann's method give positive joint power?

When the joint moment and relative angular velocity are the same sign. The muscle is driving the movement — generating mechanical energy concentrically.

When does Elftmann's method give negative joint power?

When the joint moment and relative angular velocity are opposite signs. The muscle is resisting the movement — absorbing mechanical energy eccentrically, acting as a brake.

How do you calculate work from Elftmann's method?

Work = integral of power over time. W = ∫Pj dt. In practice with discrete data: sum power × time interval across all frames.

Why is Elftmann's method useful for sport?

It reveals proximal-to-distal energy flow in throwing and kicking — showing which joints generate energy, which transfer it, and the timing of each. Directly useful for technique analysis.

What is non-compensated work?

E_total = ΣEi. Every segment's energy change is summed independently. Treats all segments as mechanically independent — energy transfer between segments is counted as if it were new work being done. Tends to overestimate total work.

What is compensated work?

ΔE_total ≠ ΣΔEi. Change in total system energy is calculated. If energy transfers freely between segments, it is not counted as work. Tends to underestimate total work by treating all transfers as free.

What is the elevator analogy for non-compensated work?

Like two independent elevators — one goes up, one goes down. They share nothing. Both movements cost energy and are counted separately. Nothing is shared.

What is the see-saw analogy for compensated work?

Like a perfect see-saw — one side goes up as the other goes down. Total system energy unchanged. No net work counted. Energy just redistributes freely.

In reality, which method should be used — compensated or non-compensated?

Neither alone. Reality sits between the two extremes. Both should be calculated and the range reported. Non-compensated = upper bound, compensated = lower bound. Truth lies somewhere between.

What does the difference between compensated and non-compensated values tell you?

How much inter-segment energy transfer is occurring. A large difference means significant energy is flowing between segments — useful information about coordination and technique efficiency.

What is hidden work?

Work produced by muscles that is completely invisible to external measurement techniques such as inverse dynamics or force plates. Cannot be measured from outside the body.

Give four examples of hidden work

1) Antagonistic co-contraction — muscles working against each other, net moment = zero but both doing work. 2) Internal tissue friction. 3) Elastic loading of ligaments and tendons. 4) Muscles moving relative to the skeleton. Also: breathing (chest pressure).

Why does hidden work mean mechanical measurements always underestimate true muscle work?

Because co-contraction, internal friction, and other hidden processes consume metabolic energy that never appears in external mechanical measurements. The energy is converted to heat invisibly.

What is apparent power?

The power measured by external techniques such as resultant joint moments and forces. Does not include hidden work.

What is the difference between external and internal power?

External power = work done on the outside world (lifting a weight, propelling a ball). Internal power = work done to move the body itself, including against gravity.

Why can a 2D scale factor not be used for 3D analysis?

A scale factor only works in a single plane. In 3D, depth information is lost in a single camera view — you cannot determine how far away an object is from a flat image alone.

What is the minimum number of cameras needed for 3D reconstruction?

Two cameras. Each provides 2 equations (u and v pixel coordinates) per digitised point, giving 4 equations to solve for 3 unknowns (x, y, z).

What is the collinearity condition?

The principle that the object point in 3D space, the camera centre, and the image point on the sensor are all collinear — they lie on the same straight line. This is the geometric basis of the DLT.

What does DLT stand for and what does it do?

Direct Linear Transformation. It mathematically maps 2D pixel coordinates (u, v) from a camera image to real-world 3D coordinates (x, y, z).

How many parameters does the DLT solve for per camera?

11 parameters (L1–L11) per camera. These describe the camera's position, orientation, and internal optical properties.

What are the 6 external camera parameters?

Position: xc, yc, zc (location of the camera in 3D space). Orientation: ψ (pan), φ (tilt), θ (roll) — the three rotation angles.

What are the 5 internal camera parameters?

Scale factors fu and fv, image centre coordinates u0 and v0, and shear factor k. These describe the camera's internal optical characteristics.

What is the minimum number of calibration points needed and why?

Minimum 6 points. Each point provides 2 equations (u and v), so 6 points give 12 equations to solve for 11 unknowns. More points = better accuracy.

Where should calibration points be placed?

Evenly spread throughout the entire capture volume — not clustered in one area. Even distribution improves reconstruction accuracy across the whole space.

What two equations does each calibration point provide?

u = (L1x + L2y + L3z + L4) / (L9x + L10y + L11z + 1) and v = (L5x + L6y + L7z + L8) / (L9x + L10y + L11z + 1). Each relates pixel position to known real-world coordinates.

How is 3D reconstruction performed?

Two cameras each provide 2 equations per digitised point (4 equations total). These are solved simultaneously using least squares to find x, y, z — the 3D real-world position.

What is the reconstruction error metric?

The RMS (root mean square) distance of the calculated (x,y,z) position from the four planes defined by the four equations. Smaller RMS = better reconstruction accuracy.

What is the synchronisation problem in 3D video analysis?

Two independently running cameras can be up to 10 ms out of phase with each other. If not corrected, the two camera views show the subject at slightly different instants, making 3D reconstruction inaccurate.

What is genlocking and what accuracy does it achieve?

A physical electronic link between cameras that forces them to trigger simultaneously. Achieves 0.000 s synchronisation error — the most accurate method.

What is LED synchronisation and what accuracy does it achieve?

A light bar visible to both cameras displays a coded pattern of illuminated LEDs. Count the lights in each camera to calculate the offset. Accuracy: approximately 0.001 s.

What is the digitised movement data synchronisation method and what accuracy does it achieve?

Vary the synchronisation offset mathematically and find the value that minimises the DLT reconstruction RMS error. The minimum RMS = correct offset. Accuracy: less than 0.001 s — the most accurate non-synchronous method.

What are the two categories of synchronisation methods?

Synchronous data methods (genlocked cameras, common event identification) and non-synchronous data methods (LED timing lights, minimising DLT reconstruction error).

Why might genlocking not always be practical?

In competition or field settings it requires a physical cable link between cameras, which may not be possible when cameras are far apart or when filming in uncontrolled environments.

How is the synchronisation offset corrected after it is found?

Interpolation is applied to one camera's data set to shift it in time by the calculated offset, producing two synchronous data sets that can then be used for 3D reconstruction.

What can 3D joint centre locations give you, and what can they NOT give you?

They CAN give you internal joint angles (using the dot product / cosine rule between vectors). They CANNOT give you full 3D rotation sequences — for that you need segment coordinate systems and more complex methods.

Why is a rugby tackle or sidestep better analysed in 3D than 2D?

These movements involve large out-of-plane components — lateral trunk lean, mediolateral foot placement, body rotation. A single 2D camera misses these entirely, making the analysis incomplete and potentially misleading.

What additional processing steps are required for 3D compared to 2D?

All standard 2D steps (smoothing, differentiation, reference frame alignment) must be repeated in three dimensions. Additionally, 3D segment axes must be defined and 3D joint rotation sequences computed — considerably more complex than 2D.

How do you calculate a 3D joint angle from joint centre coordinates?

1) Calculate vectors from the joint of interest to each adjacent joint. 2) Calculate the dot product of the two vectors. 3) Calculate the magnitude of each vector. 4) cosθ = (a·b) / (|a||b|). 5) θ = cos⁻¹(result).

What is the dot product formula for two 3D vectors a and b?

a·b = ax·bx + ay·by + az·bz. Multiply corresponding components and sum them. Result is a scalar (single number).

What is the magnitude formula for a 3D vector?

|a| = √(ax² + ay² + az²). Square each component, sum them, take the square root.

What standard video frame rate creates the synchronisation problem and what is the maximum offset?

Standard video runs at 50 frames per second (one frame every 20 ms). Two cameras can be offset by up to one full frame = maximum 10 ms synchronisation error.