1. Edward Muybridge 1830-1904 Locomotor kinematics = quantification of limb and body motion Locomotor dynamics = analysis of forces and moments Lecture # 16: Running and Walking

2. Walking = Inverted pendulum frictionless pendulum swings forever exchange kinetic & potential energy kinetic gravitational potential one stride

3. Walking v y yy  h, height of center of mass h mgh (GPE) v ½ mv 2 (KE) stride 1stride 2 G.P.E. & K.E. are out of phase (exchange possible) Walking 1)No aerial phase 2)exchange of G.P.E. and K.E.

4. Simulated Biped Robotic Biped

7. Walking What keeps mass following this arc? Walking v0v0 L required centripetal acceleration = v 0 2 /L Only source of downward acceleration is g. Therefore, mass follows arc if g > v 0 2 /L Only source of downward acceleration is g. Therefore, mass follows arc if g > v 0 2 /L…. …or v 0 2 /gL < 1 v 0 2 /gL is called Froude number (a dimensionless #), also defined as ratio of inertial to gravitational forces. DETERMINES WALK-TO-RUN TRANSITION!

8. Walking v0v0 L Example 1: Does this work for humans? What is predicted walk-to-run transition speed? V = (gL) -½ L = 0.9 m g = 9.8 m/s 2 V = 3 m/s – ‘jogging’ speed for most people Example 2: What is walk-to-run transition on moon? V = (gL) -½ L = 0.9 m g = 0.17 x 9.8 = 1.6 m/s 2 V = 1.2 m/s – ‘jogging’ speed on moon Example 3: What is walk-to-run transition for ant? V = (gL) -½ L = 0.001 m g = 9.8 m/s 2 V = 0.1 m/s ~ 100 body lengths for humans ~ 3 body lengths

9. Running = spring mass model swingstanceswingstance

10. Running swingstance height, h L -y x dx/dt = constant U vert dy/dt h GPE -k  L EPE KE E.P.E. & K.E. are partially out of phase (exchange possible) ballistic KE-EPE exchange KE-EPE exchange

11. structural element (bone) tensile element (ligament /tendon) elastic element (tendon) actuator (muscle)

13. Running Gaits Walk d.f. >0.5 Left Right stance swing stance stance+swing duty factor = Run d.f. < 0.5 Left Right aerial phase Hop 0< d.f.<1 Left Right

14. Quadrapedal Gaits stance stance+swing duty factor = Walk Left Front Left Hind Right Front Right Hind always 3 feet on ground Run Left Front Left Hind Right Front Right Hind aerial phase (trot)

15. Run Left Front Left Hind Right Front Right Hind aerial phase (pace) Run Left Front Left Hind Right Front Right Hind aerial phase (bound)

16. Run Left Front Left Hind Right Front Right Hind aerial phase (gallop)

17. COM Left Front Left Hind Left Middle Right Front Right Hind Right Middle Hexapedal Gait = Tripod