Why is mathematics important in prosthetic gait analysis? Because walking is a moving system of positions, velocities, forces and moments. Cameras estimate where body segments are; force plates measure how the ground acts on the person; models combine those observations to estimate joint loading and power. Without coordinates, vectors, trigonometry, calculus and uncertainty, a colourful gait-lab animation would be a picture rather than quantitative evidence.
This article is educational, not clinical advice. Prosthetic prescription, alignment, rehabilitation and safety belong to qualified clinicians and the individual, using current evidence and personal goals. Mathematical models simplify the body and cannot decide what is comfortable, meaningful or best for a person. Our aim is to understand how measurements support careful questions.
Quick Reading Route
- Begin with what a gait laboratory measures.
- Use coordinates and segment geometry for the motion side.
- Use ground-reaction forces for the force side.
- Work through inverse dynamics for the central mechanism.
- Read prosthetic interpretation before drawing conclusions.
- Finish with a student investigation studio and FAQs.
What a Gait Laboratory Measures
A modern laboratory may combine motion-capture cameras, force plates, video, electromyography and metabolic measurements. The US Department of Veterans Affairs Motion Study Laboratory describes high-speed cameras, embedded platforms for ground-reaction forces and moments, EMG equipment and metabolic analysis among its instruments. Each system observes a different part of the walking problem.
Motion capture estimates kinematics: position, orientation, velocity and acceleration. Force plates record kinetics: forces and moments at the foot–ground interface. EMG records electrical activity associated with muscle activation but is not a direct force meter. Metabolic instruments estimate energy-related physiological quantities. Combining these sources requires synchronised clocks, shared coordinates and explicit assumptions.
Did You Know? A marker is not a bone
Reflective markers are attached to skin or prosthetic components, while a mathematical model estimates the underlying segment pose. Skin can move relative to bone, markers can be hidden, and placement can vary. The computer animation therefore represents a model fitted to measurements, not a transparent view inside the body. This distinction is central to responsible data science.
Events divide the gait cycle
A gait cycle can be described from one event, such as initial contact of a foot, to the next corresponding event for that foot. Time may be normalised to 0%–100% so trials of different duration can be compared. Normalisation aligns phases, but it also hides absolute duration unless cadence and stride time are reported separately.
| Quantity | Example unit | What it describes |
|---|---|---|
| Step length | m | distance between successive contacts of opposite feet |
| Stride length | m | distance between successive contacts of the same foot |
| Cadence | steps/min | step frequency |
| Walking speed | m/s | distance travelled per time |
| Joint angle | degrees or radians | relative orientation of adjacent segments |
| Ground-reaction force | N | force of the ground on the person |
| Joint moment | N·m | estimated rotational effect about a joint |
| Joint power | W | rate of mechanical energy transfer |
Coordinates, Segments and Joint Angles
Choose a laboratory coordinate system: for example, x forward, y sideways and z upward. A marker position becomes a vector r = (x,y,z). If the marker moves from (0.40, 0.12, 0.88) m to (0.43, 0.11, 0.90) m in 0.02 s, its average velocity is (1.5, −0.5, 1.0) m/s. The components say more than speed alone because they preserve direction.
The magnitude of this velocity is √(1.5²+0.5²+1.0²) ≈ 1.87 m/s. Notice that the negative sideways component contributes positively to magnitude after squaring. A vector can point left while its magnitude remains non-negative.
Segment vectors
In a two-dimensional teaching model, let the knee be K = (0.10,0.55) m and ankle A = (0.18,0.12) m, using forward and vertical coordinates. The shank vector from knee to ankle is A−K = (0.08,−0.43) m. Its length is √(0.08²+0.43²) ≈ 0.437 m.
The segment orientation relative to the forward axis can be found with atan2(−0.43,0.08). Using atan2 rather than an ordinary arctangent preserves the correct quadrant. Software conventions differ, so a report must state axis directions, angle sign and anatomical definitions.
Joint angles are relative
A knee angle is derived from the relative orientation of thigh and shank segments, not from one segment alone. If the thigh orientation is −75° and the shank orientation is −100° under a chosen convention, the relative difference is 25°. Another laboratory could use a different zero and sign, yielding a different numeric display for the same pose. Definitions come before comparison.
Three-dimensional rotations do not commute
In 3D, rotating about x and then y generally produces a different orientation from rotating about y and then x. Euler-angle sequences must therefore be stated. This is not a mathematical nuisance; it explains why apparently similar angle labels from different modelling conventions may not be directly comparable.
Velocity, Acceleration and Filtering
Velocity is the rate of change of position, and acceleration is the rate of change of velocity. Motion-capture data are sampled, so derivatives are estimated numerically. For equally spaced times, a central velocity estimate is [x(t+h)−x(t−h)]/(2h). A central acceleration estimate is [x(t+h)−2x(t)+x(t−h)]/h².
The second derivative is especially sensitive to noise because small coordinate errors are divided by h². If h is 0.01 s, h² is 0.0001 s². A tiny position fluctuation can become a conspicuous acceleration spike. Filtering is necessary in many analyses, but filter choice affects peaks and timing.
A transparent classroom example
Suppose a marker’s forward coordinates at 0.98, 1.00 and 1.02 s are 0.742, 0.760 and 0.781 m. The central velocity at 1.00 s is (0.781−0.742)/0.04 = 0.975 m/s. The central acceleration is (0.781−2×0.760+0.742)/0.0004 = 7.5 m/s². If the middle coordinate were 0.761 m, acceleration would become 2.5 m/s². A 1 mm change altered the estimate by 5 m/s².
This sensitivity teaches two habits: keep raw data, and document processing. A graph without its sampling rate and filter description is incomplete evidence.
Synchronisation
Imagine force data are delayed by 0.02 s relative to marker data. At 100 Hz, that is two motion frames. The model could pair a force peak with the wrong pose, shifting estimated joint moments. Laboratories use shared triggers and time bases because inverse dynamics depends on matching motion and force at the same instant.
Ground-Reaction Forces as Vectors
Newton’s third-law description distinguishes the force of the foot on the ground from the force of the ground on the foot. A force plate reports the latter after calibration and coordinate transformation. In three dimensions, F = (Fx,Fy,Fz), often with vertical, forward–backward and side-to-side components under the laboratory convention.
If a fictional sample gives F = (−80, 25, 720) N, the magnitude is √(80²+25²+720²) ≈ 724.9 N. Reporting only the magnitude loses direction. The −80 N forward-axis component may represent braking under one convention; later a positive component may represent propulsion.
Normalising force by body weight
For a person of mass 70 kg, body weight is approximately mg = 70×9.81 = 686.7 N. A vertical force of 720 N is 720/686.7 ≈ 1.049 body weights. Normalisation supports comparison across body size, but it must be labelled. “1.05 BW” is not 1.05 N.
For a prosthesis user, analysts must be clear about whether reported mass includes the prosthesis and how segment parameters were assigned. Normalisation is a modelling decision as well as arithmetic.
Centre of pressure and moment arm
The force plate also supports estimation of a centre of pressure, the effective application point of the resultant force on the plate surface. If the vector from a joint centre to that point is r, the external moment is r × F. In a simple 2D case, moment about the joint is M = rₓF_z − r_zFₓ.
Let r = (0.10,−0.45) m from knee to centre of pressure and F = (−80,720) N. Then M = 0.10×720 − (−0.45×−80) = 72−36 = 36 N·m under the chosen sign convention. Both force direction and application point matter. Moving the same force changes the moment.
Impulse
Impulse is the integral of force over time. Numerically it is the area under a force–time curve. If average forward force is −60 N for 0.12 s, the braking impulse is −7.2 N·s. A later average of +45 N for 0.16 s gives +7.2 N·s. These balance in this simplified example, though real walking also involves changing momentum, slope and measurement details.
Inverse Dynamics: Building from the Ground Up
Forward dynamics begins with forces and predicts motion. Inverse dynamics begins with measured motion and external forces, then estimates net joint forces and moments consistent with Newton–Euler equations. “Inverse” does not mean reversing time; it describes solving for causes from observed consequences within a model.
For a rigid segment, translational balance is ΣF = ma. Rotational balance about the centre of mass is ΣM = Iα, where I is moment of inertia and α angular acceleration. Starting with the foot, an analyst uses the ground-reaction force, segment weight and measured acceleration to estimate the ankle reaction. That result becomes an input to the shank equation, continuing toward the knee and hip.
One-segment teaching calculation
Consider a fictional shank–prosthetic assembly of mass 3.8 kg with upward centre-of-mass acceleration 0.50 m/s². Suppose an upward ankle reaction of 310 N acts at the distal end. Taking upward as positive, vertical force balance gives knee reaction + 310 − 3.8×9.81 = 3.8×0.50. Therefore knee reaction = 1.9 + 37.278 − 310 ≈ −270.8 N. The negative result means downward under the chosen sign convention.
This number is not a measured knee contact force. It is a net intersegmental reaction in an intentionally simplified model. Muscles, ligaments and contact forces combine internally, and co-contraction can be substantial. Inverse dynamics does not uniquely split a net moment among individual tissues.
Net joint moment
Rotational balance includes the moments of external forces, gravity, distal reactions and the unknown proximal moment. Segment length, centre-of-mass location, inertia and angular acceleration all matter. If an assumed centre of mass shifts, moment arms change. If marker noise changes angular acceleration, the inertial term changes. This is why segment-parameter uncertainty deserves attention.
Joint power
Net joint power is commonly calculated as P = Mω, the product of net joint moment and angular velocity using consistent sign conventions. If M = 42 N·m and ω = 1.8 rad/s, P = 75.6 W. Positive and negative power are often interpreted as net generation or absorption within the model convention, not as a direct meter of one muscle’s work.
Mechanical work over an interval is the integral of power with respect to time. Positive and negative work should usually be distinguished because simply summing signed values can hide large exchanges that cancel.
Worked Comparison of Two Fictional Prosthetic Conditions
Suppose a laboratory studies two alignment conditions at a comfortable self-selected speed. The following values are invented for teaching.
| Measure | Condition A | Condition B |
|---|---|---|
| Speed | 1.05 m/s | 1.08 m/s |
| Prosthetic-side stance time | 0.66 s | 0.64 s |
| Peak vertical force | 1.04 BW | 1.07 BW |
| Peak knee net moment | 0.42 N·m/kg | 0.39 N·m/kg |
| Step-width variability | 0.021 m | 0.018 m |
| Comfort rating | 7/10 | 8/10 |
Condition B has a slightly faster speed. That alone can change forces and moments, so differences should not be attributed automatically to alignment. A matched-speed comparison, repeated trials or a statistical model could help separate effects.
Absolute and relative differences
The knee-moment difference is −0.03 N·m/kg, about −7.1% relative to 0.42. The comfort difference is one point on an ordinal rating. It would be misleading to say comfort improved by 14.3% merely because 1/7 ≈ 14.3%; rating scales do not necessarily have a meaningful ratio zero or equal perceptual intervals.
Variability and sample size
If step width is calculated from only three steps, the variability estimate is unstable. Ten or twenty usable steps may tell a different story, depending on protocol. Multiple strides from one person are not equivalent to independent participants. Statistical independence and repeated-measures structure matter.
Multicriteria reasoning
No single column decides the preferred condition. Comfort, stability, terrain, daily activities, skin health, energy cost, reliability and personal priorities can matter. Mathematics organises trade-offs, but weights should be discussed rather than hidden inside a mysterious score.
What Prosthetic Gait Analysis Can and Cannot Say
The VA Research report on a Gait and Motion Analysis Lab describes force plates that measure ground-reaction forces and a research aim of building evidence about prosthetic choices. That illustrates a proper role for gait analysis: quantify outcomes, compare conditions and contribute evidence. It does not imply that one instrument automatically selects the right limb for everyone.
It can quantify patterns
Gait analysis can describe symmetry, timing, joint kinematics, external forces, net moments, power and variability under defined test conditions. It can reveal whether a change is consistent across repeated trials and whether an apparent visual difference has measurable support.
It cannot observe every internal force directly
Inverse dynamics produces net quantities from a model. It cannot uniquely determine each muscle force or socket pressure without additional models and sensors. Even musculoskeletal optimisation requires assumptions about objectives and physiology.
Laboratory walking is a sample
A few clean steps on a flat walkway may not represent rain, crowds, slopes, fatigue or home environments. Wearable sensors and community studies can broaden evidence, but they introduce their own calibration, drift and context problems.
The person is not an average curve
Group averages can hide individual responses. An intervention with a favourable mean may not suit every participant. Plotting each person’s paired change alongside the mean helps preserve individuality.
Uncertainty, Sensitivity and Model Limits
Uncertainty enters through marker placement, skin motion, force calibration, event detection, filtering, segment mass estimates, joint-centre calculations and trial variability. A useful uncertainty budget names sources rather than attaching one unexplained ± value.
Sensitivity analysis
Recalculate a result after shifting a joint centre by 5 mm, changing a cutoff frequency within a defensible range or varying a segment parameter. If the conclusion reverses easily, report it as fragile. If it remains similar, confidence in the qualitative pattern increases.
Missing markers
Interpolation can fill a short gap, but the method assumes a trajectory shape. Linear interpolation draws a straight path between endpoints; spline interpolation imposes smoothness. Long gaps during rapid movement are harder to justify. A reconstructed point should be flagged, not quietly treated as measured.
Multiple comparisons
A gait report may contain hundreds of time points and many variables. Searching them all for the largest difference increases the chance of a striking result by coincidence. Pre-specified outcomes, confidence intervals and appropriate statistical methods help limit selective storytelling.
Statistical significance and practical importance
A very consistent 1° angle change might be statistically detectable yet unimportant for the person. A meaningful comfort improvement in a small study might not meet a conventional p-value threshold. Effect size, interval, context and patient goals belong together.
A Student Investigation Studio
Use only fictional data, toy models or publicly available de-identified teaching datasets. Do not use exercises to adjust a real prosthesis.
Investigation 1: Coordinate reconstruction
Plot hip, knee and ankle points for ten frames. Calculate thigh and shank lengths. Because rigid segments should have nearly constant length, use variation as a marker-quality check. Explain why perfect constancy is unrealistic with skin markers.
Investigation 2: Angle conventions
Calculate the same knee angle using two sign conventions. Write a definition beside each result. Show that disagreement in sign does not necessarily mean disagreement in pose.
Investigation 3: Differentiate noisy motion
Create a smooth sinusoidal position signal, sample it at 100 Hz and add small random noise. Estimate velocity and acceleration. Compare raw and gently smoothed results. State which peaks move after filtering.
Investigation 4: Force vector decomposition
For five fictional force vectors, calculate magnitudes, body-weight-normalised vertical components and forward impulses. Draw arrows to scale. Explain why equal magnitudes can have different functional directions.
Investigation 5: Moment arms
Hold a 500 N force constant while moving its application point from 5 cm to 15 cm from a joint. Plot moment against distance. The linear relationship makes leverage visible.
Investigation 6: Synchronisation error
Create angle and force curves with peaks 0.10 s apart. Shift one series by 0.02 s and calculate Mω products. Quantify the change in peak power. Relate the result to clock alignment.
Investigation 7: Inverse-dynamics balance
Draw a free-body diagram for a single shank. Label weight, distal force, proximal force, centre of mass and moment arms. Write equations before inserting numbers. Use signs consistently and check newtons versus newton-metres.
Investigation 8: Normalisation choices
Compare the same 700 N vertical force for masses of 55, 70 and 90 kg. Express each in body weights. Then discuss what normalisation reveals and what it conceals.
Investigation 9: Symmetry indices
Compute difference, ratio and a symmetric percentage for left and right step lengths. Swap left and right labels and observe which metric changes sign or reciprocal form. Choose a metric whose behaviour matches the question.
Investigation 10: Repeated measures
Create data for six fictional participants under conditions A and B. Plot paired lines, not only group bars. Calculate each change and the mean change. Identify whether everyone responds in the same direction.
Investigation 11: Decision matrix
Score comfort, stability, speed and effort for three fictional options. Try two sets of weights. If the ranking changes, explain that the choice depends on values as well as measurements.
Investigation 12: Reproducible report
Document coordinate axes, sampling rates, filters, event definitions, normalisation, excluded trials and formulas. Ask another student to reproduce one plot. Treat discrepancies as opportunities to improve the method.
Misconceptions Worth Correcting
“Symmetry is always the goal”
Symmetry can be informative, but perfect numerical symmetry is not automatically optimal, attainable or meaningful. Safety, comfort and individual anatomy matter. A symmetry index should answer a defined question, not become a universal score.
“The force plate measures joint force”
It measures external forces and moments at the ground interface. Joint quantities are estimated through equations and models. Internal contact and muscle forces are not identical to net inverse-dynamics results.
“More cameras remove uncertainty”
Additional views can reduce occlusion and improve reconstruction, but marker placement, calibration and model assumptions remain. Quantity of data does not replace data quality.
“A computer model is objective”
Computation is reproducible when inputs and rules are fixed. Choosing markers, segment definitions, filters and outcomes still involves judgement. Objectivity grows through transparency, validation and sensitivity checks.
“Faster is always better”
Speed is one outcome. A change that raises speed may also change stability, energy cost or comfort. Responsible evaluation keeps multiple goals visible.
Learning Pathways for Singapore Students
Lower-secondary mathematics supports coordinates, scale drawing, averages and speed. Upper-secondary mathematics adds trigonometry, vectors, functions and statistics. Additional Mathematics introduces differentiation and integration. Physics supplies forces, moments, energy and momentum. Computing adds data cleaning, synchronisation and visualisation.
Students interested in biomedical engineering can practise translating between a physical sketch, an equation, code and a graph. Students interested in healthcare can focus on measurement validity and communication. Students interested in design can explore how user priorities constrain optimisation. No school subject guarantees a career, but strong mathematics expands the questions a student can investigate.
A four-week plan
- Week 1: coordinates, vectors, segment lengths and angle conventions;
- Week 2: velocity, acceleration, filtering and event timing;
- Week 3: forces, moments, impulse, inverse dynamics and power; and
- Week 4: paired comparisons, uncertainty, ethics and a reproducible mini-report.
Parents can encourage careful language: “estimated joint moment” rather than “measured knee force”; “fictional condition” rather than “best prosthesis”; “association” rather than “cause.” Precision in words protects precision in reasoning.
Frequently Asked Questions
What is inverse dynamics in one sentence?
It uses measured motion, external forces and a mechanical model to estimate net forces and moments that are consistent with the observed movement.
Why are force plates embedded in the floor?
They measure the forces and moments transmitted through the foot–ground contact while allowing the person to walk across a level surface. A clean foot strike on the plate is important for assigning the measurement.
Is ground-reaction force equal to body weight?
Not throughout walking. Body weight is mg, while ground-reaction force changes as the body accelerates and as support transfers. Its time average and components depend on the analysed interval and motion.
Why use radians in power calculations?
Angular velocity in rad/s combines naturally with moment in N·m to give watts. Degrees per second must be converted by multiplying by π/180.
Can joint power identify a particular muscle?
No. It is a net model quantity at the joint. Multiple muscles and passive structures can contribute, and co-contraction is not uniquely resolved.
Why normalise moments by body mass?
N·m/kg can make comparisons across body sizes more interpretable. It does not remove all anatomical differences, and the normalisation method must be stated.
What is a gait-cycle percentage?
It maps an interval between repeated events to 0%–100%. This aligns phases across strides but should not replace reporting actual time when duration matters.
Why can marker placement change results?
Markers help define segment axes and joint centres. A placement shift changes those estimated geometries, which can alter angles, moment arms and inverse-dynamics outputs.
Does a statistically significant difference choose a prosthesis?
No. It describes evidence under a statistical model. Clinical relevance, uncertainty, comfort, activity goals, safety and individual response remain essential.
Why plot individual participants?
An average can conceal opposite or highly varied responses. Paired individual plots show who changed, by how much and in which direction.
Can students build a gait lab with phone video?
They can explore simple timing and 2D coordinate ideas with safe, consented demonstrations, but should not claim clinical accuracy. Perspective, frame rate, lens distortion and marker placement limit results.
What is the most transferable skill here?
Building a chain from measurement to model to conclusion while labelling assumptions. That skill applies to robotics, sports, animation, transport and experimental science.
Useful Next Reading
Extended case study: the result changes when the clock slips
A fictional gait laboratory compares two prosthetic settings. Cameras sample at 100 Hz and the force plates at 1,000 Hz. The acquisition computer is supposed to align both streams to one trigger. During audit, a student notices that the vertical-force rise begins two motion frames before the foot marker reaches the plate. That 0.02 s discrepancy is small to a casual viewer but large enough to pair the force with the wrong segment acceleration.
The team first resamples the force signal to the motion timeline using a documented method. They calculate ankle net moment with the original alignment, then with force shifted by one and two frames. Peak moment changes by 4%, 9% and 15% across different strides. This sensitivity shows that reporting one peak to three decimal places would be misleading. The time-alignment uncertainty dominates the last digits.
They then compare joint power. Because power multiplies moment by angular velocity, misalignment can combine a moment from one phase with angular velocity from another. A positive peak moves and briefly changes sign. The team refuses to label the sign change “energy generation” until synchronisation is resolved. This is an example of mathematics preventing an attractive but unsupported story.
After checking the hardware log, they find that one stream was exported with a different trigger offset. Correcting it brings the foot contact, force onset and video into agreement. The audit record preserves both versions, the correction rule and the reason. Reproducibility means another analyst can reconstruct what changed.
Extended case study: averages conceal responders
Six fictional participants try settings A and B. Changes in comfortable speed are +0.12, +0.09, +0.05, −0.01, −0.08 and −0.11 m/s. The mean change is +0.01 m/s, which sounds like “no difference.” Yet the paired plot shows three people faster, two slower and one nearly unchanged. The mean answers a group-average question but not “Who responded?”
The team adds comfort changes: +1, +2, 0, +2, +1 and −1 points on a ten-point scale. Speed and comfort do not move together for everyone. A weighted composite score produces different rankings when comfort weight changes from 40% to 70%. Rather than hide this sensitivity, the report displays both weight sets and invites clinical discussion.
The students also avoid six separate claims of success or failure. With very few observations per person and natural stride variation, apparent changes may not be stable. They calculate within-condition variability, show confidence intervals where justified and label this as exploratory evidence. A prosthetic decision is shared, individual and broader than one laboratory outcome.
Free-body diagrams as an error-checking language
Before touching equations, draw each segment separately. Mark the centre of mass, gravity, proximal and distal reactions, external forces and coordinate axes. Every force in a translational equation must have newtons; every term in a moment equation must have newton-metres. If a force appears without an application point in a moment balance, the diagram is incomplete.
Check limiting cases. With zero acceleration, forces should reduce to static balance. With zero angular acceleration, net moments must balance. Reversing an axis should reverse relevant component signs but not physical magnitudes. These tests catch code errors more reliably than staring at a smooth curve.
Ethical quantitative design
Gait data can be personally sensitive. A responsible student project uses consented, de-identified or synthetic data; collects only what is necessary; and avoids sharing recognisable video. Accessibility also matters: a lab protocol should not treat every participant as able to walk the same number of trials or at a prescribed speed. Missing data may reflect fatigue or safety, not carelessness.
Mathematics is ethical when it keeps the person’s goals visible, quantifies uncertainty and makes model assumptions challengeable. It becomes harmful when a score is used to overrule lived experience or when a group average is presented as destiny for an individual.
Model validation is a continuing question
An inverse-dynamics program can satisfy its equations and still represent anatomy imperfectly. Validation asks whether outputs agree with independent evidence strongly enough for the intended use. Segment lengths can be checked against measured distances; reconstructed markers can be compared with held-out observations; force integration can be compared with expected momentum change; and calculated ground contact can be checked against video.
No single check validates every output. A model accurate for step timing may be inadequate for socket-load estimation. A two-dimensional model may teach sagittal-plane mechanics while missing important transverse motion. Intended purpose defines what “good enough” means.
Students should report validation failures as results. If a reconstructed foot penetrates the virtual floor, do not crop the animation. Check coordinate transformations, marker definitions and calibration. If left–right conventions reverse after export, correct the pipeline and document it. These habits turn debugging into scientific reasoning.
A compact checklist before comparing conditions
- Were trials collected under comparable speed and assistance?
- Are the same coordinate, filtering and event rules used?
- Are forces assigned to the correct foot and time?
- Are normalisations and units identical?
- Are enough usable strides included, with exclusions explained?
- Are individual paired changes visible beside averages?
- Does the conclusion stay within laboratory conditions and model limits?
A comparison that passes this checklist is not automatically clinically decisive, but it is much harder to misread. Good mathematics creates a clear path from raw measurement to a modest conclusion.
Compare body motion with Why Mathematics? | Sports Statistics, Speed and Performance, where rates and fair comparisons matter. Why Mathematics? | Railway Wheel Conicity, Hunting Oscillation and Ride Stability shows another moving system where geometry, dynamics and stability interact. Why Mathematics? | Comparing Percentages Fairly is useful before interpreting normalised differences.
Mathematics does not reduce a person to a graph. Used well, it does the opposite: it makes assumptions visible, preserves individual responses and supports better questions. That is why gait analysis is a powerful lesson in humane quantitative reasoning.