Mechanics of Legged Locomotion. Running. Dr Dan Dudek

Transcription

1 Mechanics of Legged Locomotion Running Dr Dan Dudek North Cross School 1

2 Wheel Hypothesis Sir James Gray 2

3 BioWheel? Stomatopod Flagella 1,000 rpm, 98% efficiency 3

4 Running and Bouncing Balls Running is like a bouncing ball. Kinetic & Potential Energy Elastic Strain Energy Kinetic & Potential Energy 4

5 Energy Storage Running is like a Pogo Stick. Kinetic Potential Energy Elastic Strain Energy Kinetic Potential Energy 5

6 Energy at COM without Storage E KE (J) Run Stance E PE (J) E TOT (J) Time (s) 6

7 Energy Storage and Return E KE + E PE (J) Run Stance E ESE (J) Elastic Strain Energy (ESE) Stored in Legs E TOT (J) Time (s) 7

8 Energy Storage and Return Kinetic energy (J) Right stance Left stance Run Stance 500 Gravitational potential energy (J) Right stance Left stance Time (s) 8

9 Diversity Enables Discovery How General is the Spring-mass Model of Running? 9

10 General Principle FOUR-Legged FOUR- Legged Kinetic Energy Lizard Potential Energy Dog SIX-Legged TWO-Legged EIGHT- Legged Cockroach Human Crab 10

11 Running Ghosts Mechanical Energy Kinetic Energy Potential Energy 0.1 s 11

12 Robot as a Physical Model Biologically Inspired Bouncing Robot Robot has simple springy legs. RHex Martin Buehler Boston Dynamics Dan Koditschek University of Pennsylvania Al Rizzi Boston Dynamics 12

13 Robot as a Physical Model Robot and cockroach have force patterns consistent with Springmass Templates 13

14 Robot as a Physical Model RHex Video on CD 14

15 Methods? Fore-aft (fore-aft) Lateral (right-left) + Normal (up-down, vertical) + 3-D force platform + 3 cameras 1000 frames/sec 15

16 Pogo-stick Model for Running SIX- Legged EIGHT- Legged Cockroach Crab Full and Tu, 1990 Blickhan and Full, 1987 TWO- Legged FOUR- Legged V e r t i c a l F o r c e B o d y W e i g h t Human Fore-aft F o r c e T i m e Cavagna et al., 1977 Dog 16

17 Ground Reaction Forces F vert (N) Left Front Right Front Left Middle Right Middle Left Hind Right Hind One Step One Step Body weight Foot or feet push against ground. Ground reacts and pushes back on the body or center of mass. F fore-aft (N) Time (sec) 17

18 Energy of the Center of Mass F vert (N) F v v v v v h h PE v v (m sec -1 ) v v (t) = F v - F mg / m dt + c 2 c 2 = 0 h (m m) E PE (µj) h(t) = v v (t) dt + c 3 c 3 = 0 E PE = m g h(t) Time (msec) 18

19 Energy of the Center of Mass F f-a (N) F f-a v f-a v f-a KE v f-a (m sec -1 ) E KEf-a (µj) v f-a (t) = F f-a / m dt + c 1 c 1 = average velocity of center of mass E KEf-a = 1/2 m [v f-a (t) ] 2 Time (msec) 19

20 Spring-Loaded Inverted Pendulum (SLIP) Model Δy Δy + l - Δl l θ Δl l - Δl k leg = peak F v / Δl cos θ = (Δy + l - Δl) / l Change in θ can change effective stiffness k leg = leg stiffness l = uncompressed leg length Δl = compressed leg length θ = landing angle 20

21 Scaling Landing Angle Landing angle (θ) of the Spring-loaded Inverted Pendulum is remarkably Landing angle (θ) constant Body mass (kg) 21

22 Scaling Leg Stiffness Peak F v increases more with mass than leg compression M 1.0 Peak F v (N) Δ l (m) M Body mass (kg) Farley et al.,

23 Scaling Leg Stiffness Leg stiffness of the Springloaded Inverted Pendulum increases with Stiffnes k leg (kn m -1 ) M 1.0 = M 0.67 M 0.33 body mass Farley et al.,

24 Leg Stiffness k rel = F mg d l l M 1.0 = M 0.0 M 1.0 M 0.33 M 0.33 k 100 Relative Stiffness rel,leg 10 Cockroach Crab Quail Hare Dog Kangaroo Mass (kg) Blickhan and Full,

25 Effect of Speed Stiffness (kn m-1) k = F / Δl Humans y Low speed 40 High speed (degrees) 30 y θ θ Speed (m s-1) Farley et al

26 Running in the Real World What happens when you use your leg spring on real surfaces that differ in stiffness? more compliant surfaces? Running on cushions Greene & McMahon 26

27 Running in the Real World Leg stiffness (kn m -1 ) Human Running Ferris, Louie and Farley (1998) Leg gets stiffer as surface is more compliant Surface stiffness (kn m -1 ) 27

28 Disability Removed Flex Foot 28

29 Disabled or Super-human? Fastest Man on No Legs Oscar Pistorius runs 100m in seconds (WR 9.58 seconds) and 400m in seconds (Olympic qualifying is 45.55, WR 43.18). Qualified for 2011 World Championships and won silver in 4x400m relay Paralympics WR holder in 100, 200, and 400 m sprints. 29

30 Disabled or Super-human? 30

31 Walking and Running Is it a disadvantage that animals don t have wheels?» Energy is exchanged, stored and returned during walking and running» Wheels are only good on roads» Difficult to make biowheels 31

32 Discussion Big Dog Big Dog is a robot built by Boston Dynamics Inc. (BDI) inspired from the design of quadrupedal mammals. Its purpose is to carry heavy loads over varied terrain. Use the data on the virtual leg stiffness for quadrupedal mammals to provide information to BDI. Stiffnes k leg (kn m -1 ) 32