---
# System prepended metadata

title: configuration space and work space and kinematics
tags: [cs 593 robotics]

---

---
title: configuration space and work space and kinematics
tags: cs 593 robotics
---

# configuration space and work space

## configuration space (c-space)
- space where robot operates
- specification of the position of every point of the robot

configuration of rigid body in the plane can be described using three variables
- two for the position and one for the orientation 

configuration of a rigid body in space can be described using six variables
- three for the position and three for the 

topology
- "shape" of the configuration space

### degrees of freedom
the number of variabels is the number of degrees of freedom of the rigid body
## work space (w-space)
- space where the end effector operates
- subset of the task space that the end-effector frame can reach

### task space
- space of positions and orientations of the end-effector frame
## c-space <-> w-space
### Geometrical
- going from c-space to w-space is **forward kinematics**
    - GO THROUGH LECTURE NOTES AND DRAW THE THINGS!!!
    - x = $l_1cos\theta_1 + l_2cos(\theta_1 + \theta_2)$
    - y = $l_1sin\theta_1 + l_2sin(\theta_1 + \theta_2)$
- going from w-space to c-space is **inverse kinematics**
    - ALSO DRAW THE THINGS!!!
    - $\theta_2$ = $arccos(\frac{x^2+y^2-l^2_1- l^2_2}{2l_1l_2})$
    - $\theta_1$ = $arctan(\frac{y}{x})$ - $arctan(\frac{l^2_2sin\theta_1}{2l_1l_2})$
### jacobian matrix
- LECTURE NOTES

## iterative approach for inverse kinematics
- inverse of jacobian is the error?
- initial guess robot config
- $\epsilon$ = $\infty$
```
while E < 0.001
    delta x <- endEffector - pose(theta1, theta2)
    J <- computeJacobian(q)
    delta qerror = Jdeltax
    q <- q - deltaqerror
    E = qerror
```
- packages
    - Track IK
- when can it fail
    - asking for an unreachable pose
    - initial guess is far from desired pose
## forward kinematics
find the position and orientation fo the reference frame attached to the end-effector given the set of joint positions

## inverse kinematics
determine the set of joint positions that achieves a desired end-effector configuration
## rigid-body motions

describe the motion of a rigid body moving in 3-dimensional physical space mathematically

rotation matrix
- 3 x 3 matrix representation for describing a frame's orientation
- exsponential coordinate representation
    - given a rotation matrix, there exists some unit vector and angle such that the rotation matrix can be obtained by rotating the identity frame about the unit vector by the angle
**assignment 2 info**
- implement inverse kinematics using Jacobian matrix
## miscellaneous info
- end effector is robot gripper

read up on before next class / questions
- can work space ever be greater than 3 dimensions
- 

