<Text-field style="Heading 1" layout="Heading 1">1. Quaternions</Text-field>

In this worksheet we will set up the equations to solve the forward displacement problem of Stewart-Gough platforms, using soma coordinates.

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A quaternion q is a 4-tuple q = (q[0],q[1],q[2],q[3]) where q[0] is a scalar and (q[1],q[2],q[3]) a vector. In this section we define five operations on quaternions.

CreateQuaternion := (s,v) -> s + v[1]*i + v[2]*j + v[3]*k:

RealPart := q -> coeff(coeff(coeff(q,i,0),j,0),k,0):

VectorPart := q -> <coeff(q,i,1),coeff(q,j,1),coeff(q,k,1)>:

Conjugate := q -> RealPart(q) - coeff(q,i,1)*i - coeff(q,j,1)*j - coeff(q,k,1)*k:

MultiplyQuaternions := proc(q,r) description `multiplies two quaternions`: local q0,r0,u,v,s,w: q0 := RealPart(q): u := VectorPart(q): r0 := RealPart(r): v := VectorPart(r): s := r0*q0 - u[1]*v[1] - u[2]*v[2] - u[3]*v[3]: w := <q0*v[1]+r0*u[1],q0*v[2]+r0*u[2],q0*v[3]+r0*u[3]>: w[1] := w[1] + (u[2]*v[3] - v[2]*u[3]): w[2] := w[2] - (u[1]*v[3] - v[1]*u[3]): w[3] := w[3] + (u[1]*v[2] - v[1]*u[2]): return CreateQuaternion(s,w): end proc:

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A position in 3-space given by 3 coordinates corresponds to a position vector. This vector along with a zero real part is used as input to CreateQuaternion.

<Text-field style="Heading 1" layout="Heading 1">2. Formulation of the Equations</Text-field>

The unknowns are the quaternions e and g. Since e represents a rotation, we use symbolic coordinates "r" to represent e as a quaternion.

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eq[1] := MultiplyQuaternions(e,Conjugate(e)) - 1;

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eq[2] := RealPart(MultiplyQuaternions(g,Conjugate(e)));

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for n from 1 to 6 do eq[2+n] := MQ(g,Cg(g)) + MQ(b[n],Cg(b[n])) + MQ(a[n],Cg(a[n])) - L[n]^2 + 2*RP( MQ(MQ(g,Cg(b[n])),Cg(e)) - MQ(MQ(g,Cg(e)),Cg(a[n])) - MQ(MQ(MQ(e,b[n]),Cg(e)),Cg(a[n])) ): end do:

Note that we exploited that MQ(e,Cg(e)) = 1.

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<Text-field style="Heading 1" layout="Heading 1">3. Writing the system to file</Text-field>

To write the polynomials into a nice format to file, we use the following:

printpoly := proc(p::polynom) description `writes p, two terms per line, using lprint`: local lp,i,s: lp := convert(p,list); for i from 1 to nops(lp) do s := convert(lp[i],string); if type(lcoeff(lp[i]),float) or type(lcoeff(lp[i]),integer) then if lcoeff(lp[i]) > 0 then printf("+%s",s); else printf("%s",s); end if; else if Im(lcoeff(lp[i])) > 0 then printf("+%s",s); else printf("%s",s); end if; end if; if (i < nops(lp)) and (i mod 2 = 0) then printf("\n"); end if; end do; printf(";\n"); end proc:

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