Rotational Motion and Walking

Equilibrium

Static and dynamic balance of forces and torques

Lever

Simplified model of muscle and bone action

Torque eq

τ = r × F

Topple force

Force needed to rotate body around feet

Forward bend

Spinal tension ≈1777 N for 70kg man

Compression

Fr ≈1978 N; φ = 1.34°

Moment arm

Distance between pivot and applied force

Running jump

Push-off force 2W → upward accel = g

Launch angle

≈19°, range ≈7.06m

Rot motion

Rotation around fixed axis or center

Angular pos

θ = s / r

Angular vel

ω = dθ/dt

Tangential vel

v = rω

Angular acc

α = dω/dt

Centripetal acc

ac = v²/r = ω²r

Centripetal force

Fc = m ac

Uniform circ

Constant speed, direction changes

Angular freq

ω = 2πf

Centrifuge

ac = 6.8×10⁴ m/s² ≈ 70000g

Cylinder ride

ω=1.98 rad/s, v=9.9 m/s

Friction force

Fs = μmv²/r

Tilt angle

tanθ = v²/(Rg) → θ=17°

Car corner

Static friction prevents sliding

Banked curve

tanθ = v²/(rg)

Uniform motion

Speed constant, direction varies

Torque

τ = Iα

Inertia

I = Σmr²

Pendulum

T = 2π√(L/g)

Spring law

F = -kx

SHM eq

x(t) = A sin(ωt)

Period

T = 1/f

Angular freq2

ω² = k/m or g/L

Hooke law

Restoring force ∝ displacement

Amplitude

Maximum displacement from equilibrium

Energy SHM

E = ½kA²

Rot energy

E = ½Iω²

Walking def

At least one foot on ground

Running def

Both feet off ground at times

Step motion

Double pendulum around pelvis

Inverted pendulum

Center of mass propels forward

Robot walk

Modeled on human gait (Honda ASIMO)

Torque joints

Measured during walking

Balance senses

Otoliths, semicircular canals, proprioception

Impact walk

Force = 1.2–1.8× body weight

ZMP

Zero Moment Point = total inertial moment = 0

ZMP control

Prevents falling via torque adjustment

Floor reaction

Force from ground supporting weight

Foot control

Placement adjusts stability

Stability

Maintained via feedback & sensors

Walking mech

Uses hip, knee, and foot joints

Energy walking

Stored in leg pendulum motion

Human gait

Controlled by neural & mechanical feedback

Robot control

Floor reaction, target ZMP, foot planting

Periodic motion

System repeats motion over time

Amplitude A

Max distance from equilibrium

Frequency f

Cycles per second (Hz)

Angular freq3

ω = 2πf

Relation fT

f = 1/T

Oscillation

Repeating motion due to restoring force

Example

Pendulum & spring oscillators