Damien Chablat

style vert vide
CNRS IRCCyN

Kinematic analysis of a class of analytic planar 3-RPR parallel manipulators

Kinematics equations:

EQ1 := x^2+y^2-rho1^2

EQ2 := x^2+2*c2^2-2*x*c2-rho2^2+y^2+(-2*c2^2+2*x*c2)*cos(phi)+2*y*c2*sin(phi)

EQ3 := 2*d3^2+2*c3^2+x^2+y^2-2*x*c3-rho3^2-2*y*d3+2*cos(phi)*c3*x-2*cos(phi)*c3^2+2*sin(phi)*d3*x-4*sin(phi)*d3*c3+2*sin(phi)*c3*y-2*cos(phi)*d3*y+2*cos(phi)*d3^2

EQ3 := 2*d3^2+2*c3^2+x^2+y^2-2*x*c3-rho3^2-2*y*d3+2*cos(phi)*c3*x-2*cos(phi)*c3^2+2*sin(phi)*d3*x-4*sin(phi)*d3*c3+2*sin(phi)*c3*y-2*cos(phi)*d3*y+2*cos(phi)*d3^2

3rd-degree characteristic polynomial for the analytic manipulators

Polynome := (-16*c2*c3^2-4*rho1^2*c2+16*c3*c2^2+4*c2*rho3^2+4*c3*rho1^2-4*c3*rho2^2)*T^3+(-4*d3*rho1^2+4*d3*rho2^2+32*c2*d3*c3-16*d3*c2^2)*T^2+(4*c3*rho1^2-4*rho1^2*c2+4*c2*rho3^2-16*c2*d3^2-4*c3*rho2^...

Polynome := (-16*c2*c3^2-4*rho1^2*c2+16*c3*c2^2+4*c2*rho3^2+4*c3*rho1^2-4*c3*rho2^2)*T^3+(-4*d3*rho1^2+4*d3*rho2^2+32*c2*d3*c3-16*d3*c2^2)*T^2+(4*c3*rho1^2-4*rho1^2*c2+4*c2*rho3^2-16*c2*d3^2-4*c3*rho2^...

Polynome := (-16*c2*c3^2-4*rho1^2*c2+16*c3*c2^2+4*c2*rho3^2+4*c3*rho1^2-4*c3*rho2^2)*T^3+(-4*d3*rho1^2+4*d3*rho2^2+32*c2*d3*c3-16*d3*c2^2)*T^2+(4*c3*rho1^2-4*rho1^2*c2+4*c2*rho3^2-16*c2*d3^2-4*c3*rho2^...

with

sin(phi) := 2*T/(1+T^2) cos(phi) := (1-T^2)/(1+T^2)

Singular configurations:

S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...

S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...
S := c3^4*rho2^8-8*c3^3*rho1^6*c2*d3^2-8*c3^2*rho1^6*rho2^2*d3^2-16*rho2^2*rho1^2*c2^3*c3^3*d3^2-16*rho2^2*c2^3*d3^4*rho1^2*c3+c2^4*rho1^8-4*rho3^2*c2^4*rho1^6+12*rho1^4*c3^2*rho2^4*d3^2+2*c3^2*rho1^8*...

Q1 := -2*c3*rho1^2*c2-2*c3*rho1*c2*rho3+c3^2*rho2^2+rho2^2*d3^2+rho1^2*c2^2-2*rho1*c2*c3*rho2-2*c2*rho3*c3*rho2+d3^2*rho1^2+2*c3^2*rho1*rho2-4*c2^2*d3^2+2*rho2*d3^2*rho1+c3^2*rho1^2+c2^2*rho3^2+2*rho1*...
Q1 := -2*c3*rho1^2*c2-2*c3*rho1*c2*rho3+c3^2*rho2^2+rho2^2*d3^2+rho1^2*c2^2-2*rho1*c2*c3*rho2-2*c2*rho3*c3*rho2+d3^2*rho1^2+2*c3^2*rho1*rho2-4*c2^2*d3^2+2*rho2*d3^2*rho1+c3^2*rho1^2+c2^2*rho3^2+2*rho1*...

Q2 := -2*c3*rho1^2*c2+2*c3*rho1*c2*rho3+c3^2*rho2^2+rho2^2*d3^2+rho1^2*c2^2+2*rho1*c2*c3*rho2-2*c2*rho3*c3*rho2+d3^2*rho1^2-2*c3^2*rho1*rho2-4*c2^2*d3^2-2*rho2*d3^2*rho1+c3^2*rho1^2+c2^2*rho3^2-2*rho1*...
Q2 := -2*c3*rho1^2*c2+2*c3*rho1*c2*rho3+c3^2*rho2^2+rho2^2*d3^2+rho1^2*c2^2+2*rho1*c2*c3*rho2-2*c2*rho3*c3*rho2+d3^2*rho1^2-2*c3^2*rho1*rho2-4*c2^2*d3^2-2*rho2*d3^2*rho1+c3^2*rho1^2+c2^2*rho3^2-2*rho1*...

Q3 := -2*c3*rho1^2*c2+2*c3*rho1*c2*rho3+c3^2*rho2^2+rho2^2*d3^2+rho1^2*c2^2-2*rho1*c2*c3*rho2+2*c2*rho3*c3*rho2+d3^2*rho1^2+2*c3^2*rho1*rho2-4*c2^2*d3^2+2*rho2*d3^2*rho1+c3^2*rho1^2+c2^2*rho3^2-2*rho1*...
Q3 := -2*c3*rho1^2*c2+2*c3*rho1*c2*rho3+c3^2*rho2^2+rho2^2*d3^2+rho1^2*c2^2-2*rho1*c2*c3*rho2+2*c2*rho3*c3*rho2+d3^2*rho1^2+2*c3^2*rho1*rho2-4*c2^2*d3^2+2*rho2*d3^2*rho1+c3^2*rho1^2+c2^2*rho3^2-2*rho1*...

Q4 := -2*c3*rho1^2*c2-2*c3*rho1*c2*rho3+c3^2*rho2^2+rho2^2*d3^2+rho1^2*c2^2+2*rho1*c2*c3*rho2+2*c2*rho3*c3*rho2+d3^2*rho1^2-2*c3^2*rho1*rho2-4*c2^2*d3^2-2*rho2*d3^2*rho1+c3^2*rho1^2+c2^2*rho3^2+2*rho1*...
Q4 := -2*c3*rho1^2*c2-2*c3*rho1*c2*rho3+c3^2*rho2^2+rho2^2*d3^2+rho1^2*c2^2+2*rho1*c2*c3*rho2+2*c2*rho3*c3*rho2+d3^2*rho1^2-2*c3^2*rho1*rho2-4*c2^2*d3^2-2*rho2*d3^2*rho1+c3^2*rho1^2+c2^2*rho3^2+2*rho1*...