Lecture 16: Forces and Energy Cost in Locomotion

Slide 1

• Bridges the muscle physiology unit (Lectures 11–15) to the whole-organism mechanics of locomotion.
• Goal: connect force and mechanical energy demands of movement to the muscular and metabolic energy demands that follow.

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Slide 2

Learning Objectives

1. Describe the basic mechanical demands of legged locomotion and how they relate to the force and mechanical work demands of bipedal gaits.
2. Describe how force demands of gait vary with speed and body size.
3. Discuss how leg posture influences the muscle force required relative to the external ground reaction force.

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Slide 3

Why Do We Move the Way We Move?

• Animals exhibit a surprisingly small number of characteristic terrestrial gaits — most can be identified from a single snapshot.
• The convergence on a few stereotyped patterns suggests that physical principles (not just anatomy or development) constrain efficient locomotion.

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Slide 4

The Ministry of Silly Walks

• Humans are not physically constrained to walk the way we do — many other patterns are mechanically possible.
• The fact that we converge on a narrow set of gaits underscores that the patterns we use are energetically favored, not anatomically forced.
• Modern AI-based gait recognition exploits the small individual signature embedded in the otherwise stereotyped pattern.

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Slide 5

Walking and Running — Same Patterns Across Species

• Despite very different morphologies, humans and ostriches show essentially the same dynamics for walking and for running.
• The schematics introduce the mass-spring model of locomotion that will recur throughout the lecture.

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Slide 6

Convergence Despite Different Development

• A young ostrich (a few weeks old) shows the same gait dynamics as the adult — only scaled in body size.
• Humans take many months and many practice steps to learn to walk, yet converge on similar mechanics as a precocial bird that walks shortly after hatching.
• Strong evidence that fundamental physical principles — not just developmental learning — drive the choice of gait.

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Slide 7

Setting Up the Physical Framework

• Transition slide. The remainder of the lecture develops the basic physical demands of legged movement and uses them to predict gait mechanics.

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Slide 8

Three Basic Demands

• Move the center of mass from point A to point B.
• Support body weight against gravity throughout each stance.
• Oscillate the legs to position them for the next stance phase.
• These three constraints are the foundation for understanding all the force and energy patterns of gait.

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Slide 9

Newton’s Laws and the Ground Reaction Force

• Vertical equilibrium across a complete stride requires:

\[\sum F_{vertical} = Mg + F_{legs} = 0 \quad \Rightarrow \quad F_{legs} = -Mg\]

• The ground reaction force (GRF) is the equal-and-opposite force the ground exerts on the foot — measured directly by a force platform.
• Averaged over an integer number of strides, vertical GRF must equal body weight; horizontal GRF must average zero at constant speed.

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Slide 10

Why Legs Are Different from Wheels

• Legs involve intermittent ground contact: each step decelerates and re-accelerates the body.
• This creates collisional energy losses at each foot-down event — a key driver of the muscular work demand explored later in the lecture.

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Slide 11

Measuring Ground Reaction Forces

• A force platform embedded in the track measures the GRF during one stance phase.
• The vertical GRF is largest in magnitude (multiple body weights at peak).
• The horizontal GRF is biphasic: negative (decelerating) at heel strike, positive (accelerating) at push-off.

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Slide 12

Multi-Plate Force Measurement Setup

• Sequential force plates allow measurement of GRF across several consecutive steps, enabling analysis of the full stride cycle and any unsteady (accelerating, perturbed) movements.
• The same paradigm is used in animal experiments and in human gait labs.

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Slide 13

The Mass-Spring Model of Bouncing Gaits

• A single point mass on a massless springy leg reproduces both the magnitude and the timing of GRF in steady locomotion.
• Holds across an enormous range of legged animals — cockroach, dog, ostrich, human — for bouncing (running, hopping, trotting) gaits.
• For multi-legged animals, all stance limbs are approximated as a single virtual leg.

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Slide 14

Walk vs. Grounded Run vs. Aerial Run

• Walking: characteristic M-shaped vertical GRF with a double-support phase (both feet on the ground).
• Grounded run (e.g., race-walking): single GRF peak, but no aerial phase — duty factor still > 0.5.
• Aerial run: tall single GRF peak with a true aerial phase between steps; peak force grows because stance duration shrinks as speed increases (Slide 27).

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Slide 15

Foot-Strike Pattern and Impact Peaks

• Rear-foot strikers (heel contact first) show a brief early impact peak in the vertical GRF — caused by the abrupt collision of a relatively rigid heel.
• Forefoot strikers show a smoother rise because the foot’s arch acts as a spring to cushion the contact.
• Rear-foot striking became common with the invention of running shoes — most barefoot running is forefoot-strike.

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Slide 16

Walk = Inverted Pendulum; Run = Bouncing Spring

• Walking: E_g and E_k are out of phase — the body vaults over a stiff leg, exchanging gravitational potential and kinetic energy like an inverted pendulum. Little muscular work needed within stance.
• Running: E_g and E_k are in phase — the body bounces on a compliant leg. The mechanical energy lost from the body is stored elastically in tendons and ligaments and returned at push-off.

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Slide 17

What Passive Cycling Cannot Eliminate

• Both inverted-pendulum (walk) and elastic-spring (run) mechanisms minimize the mechanical work the muscles must do within the stance phase.
• They cannot eliminate two demands:
  1. Force demands to support body weight.
  2. Energy losses at step-to-step transitions (collisions when the next foot lands).
• These two unavoidable demands set the minimum muscular work of locomotion.

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Slide 18

Prosthetic Limbs as Mass-Spring Systems

• The mass-spring model of running motivates the design of carbon-fiber running blades for athletes with transtibial amputations.
• Three blade designs (1E90 Sprinter, Catapult FX6, Cheetah Xtend) have very different stiffnesses, which can be tested directly against the model.

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Slide 19

Lower Prosthetic Stiffness Lowers Running Cost

• Across three blade types, stiffer prostheses raise the net cost of transport for running.
• A more compliant blade allows greater elastic energy cycling, mimicking the natural Achilles tendon’s energy-storage function.
• A direct application of the mass-spring framework: simple physics-based models can guide assistive-device design.

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Slide 20

The Collisional Perspective

• During the double-support phase of walking, the trailing leg pushes off while the leading leg collides with the ground.
• Push-off adds mechanical energy; the collision dissipates energy.
• The amount of push-off work needed is directly proportional to the collisional energy loss — making collisions the central determinant of muscular work in walking.

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Slide 21

Collision Cost Scales as Step Length to the Fourth Power

• Center-of-mass work scales roughly as step length to the fourth power, with R² = 0.96.
• As walking speed (and step length) increases, the collisional cost explodes — a major reason humans switch to running at higher speeds, where elastic mechanisms can offset some of the cost.

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Slide 22

Rimless Wheel — A Mechanical Demonstration of Collisions

• A rimless wheel with n spikes rolling down a fixed slope is a simple physical model of legged walking.
• With fewer spokes (longer effective step length), more energy is lost at each collision and the wheel rolls slower.
• With more spokes (shorter steps), collisions are smaller and the wheel rolls faster.
• The slope is constant, so this isolates the effect of collision geometry on speed.

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Slide 23

Two Ways to Reduce Collisional Cost

• Ankle push-off by the trailing leg just before heel strike of the leading leg reverses some of the body’s downward velocity, shrinking the upcoming collision.
• Foot rolling during stance — the human foot’s plantar geometry acts like a wheel, translating the center of pressure smoothly from heel to toe.
• A demonstration with two seven-sided polygons (one with concave sides, one with convex sides) shows that even small convex curvature on the contact surface dramatically reduces collisional losses — a simple mechanical explanation for why human feet are large and curved (see also Adamczyk, Collins & Kuo studies of foot curvature).

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Slide 24

Passive-Dynamic Walking Robots

• Robots designed around the same principles — minimal actuation, rolling foot contact, ankle push-off, locked knees during mid-stance — can walk reasonably well on essentially no power.
• McGeer’s classic passive walkers descend a slope powered only by gravity.
• Concrete demonstration that the physics of collisions and elastic cycling are the dominant determinants of bipedal walking — not active neural control.

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Slide 25

Transition — Forces and Speed

• Transition into the next section: how GRF magnitudes scale with running speed.

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Slide 26

Acceleration vs. Steady Speed

• During the acceleration phase, the fore-aft GRF is net positive at every step — the runner is adding energy.
• As the runner reaches a steady speed, the fore-aft GRF shifts to its standard biphasic pattern (negative then positive, net zero).
• Posture also shifts — a forward lean during acceleration straightens out at steady speed.

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Slide 27

Why Peak Force Grows with Speed

• Average vertical GRF must equal body weight across each step.
• As speed rises, the stance time t_c shrinks and the aerial time grows — so the peak vertical force must rise to keep the time-integrated average constant.
• Roughly: peak GRF $\propto 1/t_c$.

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Slide 28

What Limits Top Running Speed

• The required average vertical GRF can be predicted from gait timing alone:

\[\dfrac{F\_{avg}}{W_b} = \dfrac{T\_{step}}{T_c} = \dfrac{L\_{step}}{L_c}\]

• Across normal running, one-legged hopping, and backward running, this relationship holds.
• Top speed is set by:
  1. The minimum achievable contact time to apply the necessary force.
  2. The fastest achievable swing frequency.
• Force demands come from physics; force limits come from muscle strength and contraction frequency.

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Slide 29

Scaling Across Body Size

• Terrestrial mammals span a ~10⁶-fold range in body mass (mouse to elephant).
• Across this enormous range, the physics of legged locomotion demands very different postural and architectural solutions — the topic of the next slides.

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Slide 30

Geometric Scaling — Why Bigger Animals Are Relatively Weaker

• Strength scales with cross-sectional area (length²).
• Body mass (and weight loading) scales with volume (length³).
• The ratio CSA / mass ∝ 1/length — so larger isometric animals are relatively weaker.
• Without compensating shape change, an elephant-sized animal would lack the structural margin to support its own weight.

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Slide 31

How Animals Solve the Scaling Problem

• Larger animals change shape rather than scaling isometrically — bones become proportionally thicker, and limb postures become more upright.
• Sets up the key scaling relationship of the next slide: effective mechanical advantage.

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Slide 32

Effective Mechanical Advantage Scales With Body Size

• Effective mechanical advantage:

\[\text{EMA} = \dfrac{r}{R}\]

• Larger animals adopt straighter limb postures that increase EMA — moving the GRF closer to the joint center, reducing the muscle force required.
• Across mammals, EMA scales positively with body mass — allowing peak muscle (and bone) forces to scale with cross-sectional area, preventing structural failure.
• (This is the same Biewener 1989 result reviewed in Lecture 14.)

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Slide 33

Crouched Posture Raises Muscular Effort

• McMahon’s classic Groucho running experiment: subjects run with deliberately flexed knees.
• Crouched posture moves the GRF moment arm R further from the joint center → larger muscle force required for the same GRF.
• Vertical GRF profiles also become smoother (reduced impact peak) — but at a steep metabolic cost.

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Slide 34

Groucho Running Costs ~50% More Energy

• Running with a deeply flexed knee raises the metabolic cost of running by ~50% above normal upright posture.
• The mechanism is exactly the lever-system equation $F_{muscle} = F_g \times R/r$ — a larger R demands a larger muscle force, which costs more ATP per stride.
• A direct demonstration that leg posture is a primary determinant of the energy cost of locomotion.

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Slide 35

Why GRF Matters

• GRF magnitude sets the muscle force demand — and hence the metabolic energy cost.
• Maximum force capacity can be performance-limiting (top speed, sharpest turn radius).
• Skeletal safety factor is typically 2–4 — peak bone stress is normally 25–50% of failure strength; unexpectedly high loads cause injury.
• Muscle force can’t be avoided, but muscle work can be minimized through passive-dynamic energy cycling (springs, pendulums).

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Slide 36

Tissue Remodeling Across the Lifespan

• Musculoskeletal tissues remodel in response to applied loads — both during training (Lecture 15) and across the lifespan.
• The MRI cross-sections compare three subjects at the same anatomical level:
  • 40-year-old triathlete: large quadriceps, minimal adipose.
  • 74-year-old sedentary: small quadriceps, extensive intramuscular and subcutaneous adipose.
  • 70-year-old triathlete: quadriceps essentially indistinguishable from the 40-year-old.
• Chronic exercise preserves lean muscle mass into older age — strong evidence for the role of mechanical loading in long-term tissue maintenance.

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Slide 37

Summary

• The regulation of GRFs is the central principle of terrestrial locomotion. Newton’s laws are unavoidable.
• Body-weight support against gravity is a fundamental demand; peak GRFs rise with speed because contact time shrinks.
• Muscle force capacity can limit running speed and turning ability.
• GRFs translate into muscle force demands through skeletal lever systems (Lecture 13–14 EMA framework).
• Passive-dynamic energy cycling (inverted pendulum, elastic spring) reduces muscle work within stance — but cannot eliminate force demands or step-to-step collisions.
• Collisions at step-to-step transitions are the dominant source of mechanical work demand; they grow with step length (~s⁴) and with speed.
• Ankle push-off and foot rolling reduce collision losses.
• Larger animals are relatively weaker by isometric scaling and compensate with straighter limb postures (higher EMA).

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Key Equations

Equation	Name	Description
$\sum F_{vertical} = Mg + F_{legs} = 0$	Vertical force balance	Average vertical force from the legs equals body weight across an integer number of strides.
$F_{avg}/W_b = T_{step}/T_c = L_{step}/L_c$	Weyand step-cycle equation	Predicts the average vertical GRF (in body weights) from the ratio of step duration to stance duration.
$F_{peak} \propto 1/t_c$	Peak force vs. contact time	As speed rises, contact time shrinks and peak vertical GRF must rise to maintain weight support.
$\text{COM work} \propto s^4$	Kuo step-length scaling	Center-of-mass work rate at step transitions scales approximately as the fourth power of step length.
$F_{muscle} = F_g \times R/r$	Limb lever equation	Muscle force needed to balance the GRF at a joint, with r = muscle moment arm and R = GRF moment arm.
$\text{EMA} = r/R$	Effective mechanical advantage	Ratio of muscle to GRF moment arms; rises with body mass as posture straightens.
Strength $\propto L^2$, Mass $\propto L^3$	Isometric scaling	Strength grows with surface area, mass with volume — larger isometric animals are relatively weaker.

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Glossary of Key Terms

Term	Definition
Center of mass (CoM)	The single point at which the body’s mass can be approximated as concentrated for whole-body dynamics.
Ground reaction force (GRF)	The force the ground exerts on the foot — equal and opposite to the force the foot exerts on the ground (Newton’s 3rd law).
Force platform	An instrumented plate that measures the three components (vertical, fore-aft, medio-lateral) of GRF in real time.
Stance phase	The portion of the stride cycle when the foot is on the ground.
Aerial phase	The portion of the stride cycle when no foot is on the ground (running, hopping).
Double support	The portion of a walking stride when both feet are on the ground.
Duty factor (DF)	Fraction of the stride cycle spent in stance; > 0.5 for walking, < 0.5 for aerial running.
Mass-spring model	A point-mass body on a massless springy leg; reproduces GRF magnitudes and timing in bouncing gaits across diverse legged animals.
Inverted pendulum model	The walking analogy in which the body vaults over a stiff stance leg, exchanging gravitational and kinetic energy out of phase.
Bouncing gait	A locomotion pattern (running, hopping, trotting) in which gravitational and kinetic energies fluctuate in phase and elastic structures cycle the energy.
Inverted pendulum	The walking mechanism: gravitational PE peaks at mid-stance while KE is minimum, allowing passive energy exchange.
Step-to-step transition	The brief interval at the end of one stance and the start of the next when the trailing leg pushes off and the leading leg collides.
Collision	The energy-dissipating impact at heel strike when the leading leg comes down with downward velocity that must be reversed.
Push-off	Trailing-leg work that adds energy to the body; effective push-off just before heel strike reduces the upcoming collision.
Foot rolling	The continuous translation of the center of pressure from heel to toe during stance, made possible by the foot’s curved geometry — reduces collision losses.
Rear-foot strike	A running foot-strike pattern (heel first) producing a small early impact peak in the vertical GRF; common in shod runners.
Forefoot strike	A running foot-strike pattern (ball of foot first) with smoother GRF rise; uses the foot arch as a spring; common in barefoot runners.
Effective mechanical advantage (EMA)	Ratio of muscle moment arm to GRF moment arm (r/R); rises with body mass and is roughly constant within a species.
Isometric scaling	Geometric scaling in which all linear dimensions grow in proportion; under isometry, strength (∝ L²) grows slower than mass (∝ L³).
Allometric scaling	Scaling with body size in which proportions change — e.g., the postural shift that raises EMA in larger mammals.
Safety factor	Ratio of failure stress to peak operating stress; bone safety factor is typically 2–4 in locomotion.
Cost of transport (CoT)	Metabolic energy required to move a unit body mass over a unit distance (J kg⁻¹ m⁻¹) — the central efficiency metric for locomotion.
Groucho running	Running with a deeply flexed knee and lowered center of mass; dramatically increases muscle force demand and metabolic cost (~50%).
Passive-dynamic walking	Bipedal locomotion driven by gravity (or minimal actuation) plus mechanical-system geometry; demonstrated in McGeer-style passive walkers.
Running blade prosthesis	Carbon-fiber prosthetic foot designed to mimic the elastic energy cycling of the natural Achilles tendon.