Commit 76fbd88f authored by Amira Abdel-Rahman's avatar Amira Abdel-Rahman
Browse files

documentation

parent 62c6cd4f
......@@ -7,10 +7,10 @@ Proposal [link.](https://docs.google.com/document/d/1dNlEJe5-szEFqpwIeSgpptBZx4W
---
## Simulation
$$
```math
q = [\theta_1 ; \theta_2; x; y]; \\
\dot{q} = [\dot{\theta}_1 ; \dot{\theta}_2; \dot{x}; \dot{y}];
$$
```
---
## Control
......@@ -29,27 +29,27 @@ Get Bezier curve path of the leg from simulation/optimization and have an Imped
### Variables
Two Bezier curves for torques:
$$
```math
ctrl_1=[T_1,T_2,..T_n] \\
ctrl_2=[T_1,T_2,..T_n] \\
n=6 \\
2<T_i<2
$$
```
Starting Conditions:
$$
```math
ground \ height = -0.164; \\
\theta_1=-36*\pi/180; \\
\theta_2=90*\pi/180;\\
x=y=0\\
\dot{\theta}_1=\dot{\theta}_2= 0\\
\dot{x}=\dot{y}= 0\\
$$
```
### Constraints
$$
```math
x^{end}>=x^{start} \\
q_1^{end}=q_1^{start} \\
q_2^{end}=q_2^{start} \\
......@@ -59,20 +59,20 @@ y^{end}=y^{start} \\
\dot{q}_2^{end}=\dot{q}_2^{start} \\
\dot{x}^{end}=\dot{x}^{start} \\
\dot{y}^{end}=\dot{y}^{start} \\
$$
```
### Objective Function
1. First objective: maximize height $h'$ to push it to go up, got the max $x'$
2. Then I added to the constraints:
$$
```math
apex \ height = h' \\
x_{end}=x' \\
$$
```
and minimized cost of transport by minimizing:
$$
```math
E/(m*g*d)
$$
```
which is the similar as minimizing sum of torque squared as m, g and d is fixed
......@@ -81,7 +81,7 @@ $$
---
## Results
###First Hop
### First Hop
<img src="./img/first_hop.gif" width="80%" /><br></br>
......
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