Wednesday, June 24, 2009
Stewart Platform
Here is an interesting offshore crane simulator made with a Stewart platform. The platform is a type of parallel manipulator that provides six degrees of freedom through a limited range.
I always wondered if you mathematically end up with a half of a degree of freedom if you have a robot that can only move forward or that can only rotate clockwise.
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3 comments:
In my opinion, "only move forward or only rotate clockwise" haven't been defined as a mathematical quantity, thus no good mathematical model can describe it consistently and simply.
As far as I know,
A lot of system define degree of freedom as the NUMBER of free variables for equation of n variables, no matter the equation is physical chemistry equation or Quaternion or translation equation with x,y,z and Euler angles. The NUMBER of free variables can not be fractional.
I think this is a fantastic question, because it challenged the limit of the existing mathematical model, you probably need to design your own mathematical system to describe the "fractional degree of freedom".
Well the idea is that mathematically theta and theta dot (rotational) or x and x dot (linear) would only be positive (ccw/fwd) or negative (cw/rev) but not both. This could have effect the results when solving inverse kinematics problems.
I can't think of any chemical whose bonds can only be twisted or pulled in only one direction. It is usually assumed that translations and rotations are reversible, however a clutch and a rachet and pawl are examples of devices that only rotate in only one direction.
A car on the highway or an airplane would be examples of systems that can not linearly reverse direction.
It appears that answer is probably related to Lie Algebra and Holonomic motion.
"Well the idea is that mathematically theta and theta dot (rotational) or x and x dot (linear) would only be positive (ccw/fwd) or negative (cw/rev) but not both. This could have effect the results when solving inverse kinematics problems.
"
If I got you right. The idea can be modeled by adding extra constraints to the equation group 1.
equation group 1:
f1(x,y,z,alpha, beta, gama)= 0
f2(x,y,z,alpha,beta,gama)=0
equation group 2:
f1(x,y,z,alpha, beta, gama)= 0
f2(x,y,z,alpha,beta,gama)=0
x>0
y>0
z>0
Diff(alpha)>0
Diff(beta)>0
Diff(gama)>0
Does the equation group 2 have fractional numbers of free variables, thus fractional degree of freedom?
I guess the "fractional degree of freedom" should be firstly defined in a mathematically describable fashion...hunn...
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