Forward Kinematics

degree of freedom

universal join 万向节

grubler’s criterion

F=λ(n−j−1)+∑i=1jfi

where, F is number of DOF,
n is number of links,
j is number of joints ,
λ is number of DOF in the space.
fi is number of DOF permitted by joint ji.

对于万向节而言， n为3， j 为2， λ 为6，
j1 为 1， j2 为1
所以万向节的自由度为2.

constrain formulation

exercise

Forward Kinematics and DH

Forward Kinematics is used to find the position of end effect.

DH Parameters

DH relies mainly on the X and Z frames

ai is distance between zi−1 and zi along xi−1.

αi is angle from zi−1 to zi along xi−1.

di is distance between planexi−1yi−1 to plane xiyi along zi−1

θ is angle about previous zi−1, from xi−1tonew x_{i}$.

xi 始终垂直并与zi相交.

DH 参数只由x，z轴确定。

exercise

function A = compute_dh_matrix(a, alpha, d, theta)


ctheta = cos(theta);
stheta = sin(theta);
salpha = sin(alpha);
calpha = cos(alpha);
A = [ctheta, -stheta*calpha, stheta*salpha, a*ctheta; stheta, ctheta*calpha, -ctheta*salpha, a*stheta; 0,salpha, calpha,d; 0,0,0,1];
end