Lecture 2

.title[
# Lecture 2
]
.subtitle[
## Terrestrial Locomotion I
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2024/1/18
]

---

```
## Warning: package 'ggplot2' was built under R version 4.3.1
```

<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.14.0/css/all.min.css">

- Consider how we characterize terrestrial walking with legs

- Focus on bipedal locomotion

- Apply the inverted pendulum model to bipedalism

- Discuss the predictions of the model

- Take a peak at AME1 and MP1

]

.pull-right[
![](https://media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fsrep13636/MediaObjects/41598_2015_Article_BFsrep13636_Fig6_HTML.jpg)
]

---

# But first, some ground rules and metrics

- **Kinematics:** Position over space and time (2D or 3D).

- **Kinetics:** Forces moving masses.

- **Dynamics:** Changes in properties like forces, mass, energy, etc. over space and time (i.e. stability).

- **Power:** How quickly work is done (i.e., energetics).

]

## Some examples

]

---

# Some ground rules and metrics

- **Kinematics:** Position over space and time (2D or 3D).

- **Kinetics:** Forces moving masses.

- **Dynamics:** Changes in properties like forces, mass, energy, etc. over space and time (i.e. stability).

- **Power:** How quickly work is done (i.e., energetics).

]

## Some examples

.center[
![](https://www.science.org/cms/10.1126/science.288.5463.100/asset/c515a8df-1d40-4ffc-a08c-a89969f4e991/assets/graphic/se1308425003.jpeg)
]

]

---

# Some ground rules and metrics

- **Kinematics:** Position over space and time (2D or 3D).

- **Kinetics:** Forces moving masses.

- **Dynamics:** Changes in properties like forces, mass, energy, etc. over space and time (i.e. stability).

- **Power:** How quickly work is done (i.e., energetics).

]

## Some examples

.center[
![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Lengthtension.jpg/400px-Lengthtension.jpg)
]

]

---

# Some ground rules and metrics

- **Kinematics:** Position over space and time (2D or 3D).

- **Kinetics:** Forces moving masses.

- **Dynamics:** Changes in properties like forces, mass, energy, etc. over space and time (i.e. stability).

- **Power:** How quickly work is done (i.e., energetics).

]

## Some examples

]

$$\textrm{Watt}= \textrm{kg}\cdot \textrm{m}^2\cdot \textrm{s}^{-3} $$

`$$1 \textrm{Watt} \approx \textrm{10.5 mL O}_2 \textrm{ min}^{-1}$$`

---
class: top
# Some ground rules and metrics

- **Stride length:** Distance between consecutive footfalls of the same foot (in m)

- **Stride frequency:** number of strides per time (if s, in Hz)

- **Duty factor:** fraction of stride time that a foot is on the ground (non-dimensional)
 
  + e.g.,human walking = 0.5 - 0.6, running = 0.35).
 
  + Gaits with duty factors less than 0.5 imply airborne phases, i.e., running

- **Relative Phase:** Point in time a foot touches as a fraction of the stride time (non dimensional or in `$^o$`, often).

]

]

---
class: top

# Up first: Walking

## First, a question

Say you're on Mars. Do you walk faster or slower than you do on Earth?

]

1/3 the gravity as Earth
]

---
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# Up first: Walking

## Modeling walking as an inverted pendulum

![](img/ip.gif)
]

---
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# What are the predictions of such a model?

- Velocity?

- Kinetic energy?

- COM oscillations (kinetics, kinematics)?

]

---
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# What are the predictions of such a model?

.pull-left[
   
![](img/pend.png)
.center[
<img src="img/centrip.png" width="150" />
`$$F_c=ma_c=\frac{mv^2}{r}$$`
]
]

Acceleration toward the pivot (center) can be expressed as

`$$a_c=\frac{v^2}{r}$$` [why?]

Which means that  `$v^2=a_cr$`, or, if `$L=r$`

`$$v^2 \leq gL$$`

[why?]

]

---
class: top

# Back to the questions

Say you're on Mars. Do you walk faster or slower than you do on Earth?

![](img/pend.png)

![](https://mars.nasa.gov/internal_resources/491)

]

`$$v^2 \leq gL$$`

then

`$$v\leq \sqrt{gL}$$`
]
]

---
class:

# How well does the model do?

Or, How well do our iverted pendula do?

`$$Fr =\frac{centripetal force}{gravitational force}=\frac{\frac{mv^2}{L}}{mg}=\frac{v^{2}}{gL}$$`
or

`$$Fr=\frac{realized force}{predicted force}$$`

]

---
class: top

# What can change speed under this model?

- Lumbar flexion
  - Pelvic rotation
  - Pelvic tilt (in transverse, frontal planes)
  - Plantar flexion
  
  More on this later . . .
  
<img src="img/pelvicwalk.png" width="200" />
]

.pull.right[

]

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# For AME1 and MP1

![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcSmBcLYF8uP0grceY4RGQz-AKoWSGxME5yBbw&usqp=CAU)
]
.pull-right[

![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTJGsX9sltpgeyrY5wI-LVcckPgYXPHCO18Aw&usqp=CAU)

IDE: Integrated Development Environment

]

- Download R first

- Download R Studio second

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# Thanks!

Slides created via the R package [**xaringan**](https://github.com/yihui/xaringan).