Dof

Coordinate System and Degrees of freedom

How do you describe the pose of a robot?

The only full description is a total coordinate system.

You can think about it with your right hand:

  • Thumb along x-axis
  • Index along y-axis
  • Middle along z-axis

Order counts.

  • Locomotion: the ability for the robot to move itself.
  • Manipulation: the ability for the robot to move other objects.

Axis-Angle or Quaternion Notation

Three vertices (x, y, z) each with 3 values.

A point is a linear combination of three coefficients with basis vectors for the 3 axis:

p = c1[1,0,0] + c2[0,1,0] + c3[0,0,1]

You can transform this into a matrix multiplication where a rotation matrix is multiplied with a vector of coefficients:

p = [
    [1, 0, 0],
    [0, 1, 0],
    [0, 0, 1]
] * [c1, c2, c3]

We only need 4 values to express rotations (9 values):

  • Each column is a basis vector
  • They are orthogonal -> the third vector is given by the other two
  • They are normal -> the third value is given by the other two

Degrees of Freedom

  • Cartesian space

An airplane can be in all possible positions and orientations - 6 DoF.

A human arm can move in 6 DoF in cartesian space.

  • Actuator Space

An airplane can pitch, roll, and accelerate along positive X - 3 DoF.

A human arm has 7 DoF in actuator space.

Kinematics

Develop equations for the translation and rotation along axis:

Using the "E Puck" as an example:

delta-x = r theta-l + r theta-r
delta-y = 0 // no wheel to move on y axis
delta-z = 0 // same...
rotate-x = 0
rotate-y = 0
rotate-z = (r theta-r - r theta-l)/d

Any time you change the configuration you have to re-think through each degree of freedom and derive the equations.

Coordinate Transforms

To understand what a position in coordinate system B means in coordinate system A, you have to take the dot product with a rotation matrix.

  • The rotation matrix consists of the basis vectors of one coordinate system expressed in the target coordinate system.
  • Constructing this element-by-element always works

Holonomy

Systems in which closed loops in joint space result in closed loops in cartesian space are holonomic.

  • all robot arms
  • robot sliding on a track

If that is not the case, they are non-holonomic

  • most wheeled platforms
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