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Maple Worksheet...

Ohmyus 15
June 25 2024
0 0

@Carl Love thank you for your tips. I've tried to follow your suggestions, and now Maple no longer stays calculating for a very long time, but instead outputs [ ] as the solution for my system. Here's the worksheet and again, thank you very much for the help.

> 

 

This document is for figuring out the transformation necessary to get the spacemouse working

> 

restart
> 

Defining the rotation matrix. T is a linearized version assuming small angles, but it's not used.
> 

L := Matrix(3, 3, {(1, 1) = cos(theta)*cos(psi), (1, 2) = cos(theta)*sin(psi), (1, 3) = -sin(theta), (2, 1) = sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi), (2, 2) = sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi), (2, 3) = sin(phi)*cos(theta), (3, 1) = cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi), (3, 2) = cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi), (3, 3) = cos(phi)*cos(theta)}); T := Matrix(3, 3, {(1, 1) = 1, (1, 2) = psi, (1, 3) = -theta, (2, 1) = phi*theta-psi, (2, 2) = phi*psi*theta+1, (2, 3) = phi, (3, 1) = phi*psi+theta, (3, 2) = psi*theta-phi, (3, 3) = 1})

Matrix(3, 3, {(1, 1) = cos(theta)*cos(psi), (1, 2) = cos(theta)*sin(psi), (1, 3) = -sin(theta), (2, 1) = sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi), (2, 2) = sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi), (2, 3) = sin(phi)*cos(theta), (3, 1) = cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi), (3, 2) = cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi), (3, 3) = cos(phi)*cos(theta)})

  

Matrix(%id = 36893490965311603164)

 (1)
> 

P_ni is the position vector of a point on the top plate before the transformation, described by a parameter rho_n
> 

p[ni] := Typesetting[delayDotProduct](R, Vector(3, {(1) = cos(rho[n]), (2) = -sin(rho[n]), (3) = 0}), true)

Vector[column](%id = 36893490965311580268)

 (2)
> 

p__n := L.p[ni]+(Vector(3, {(1) = u, (2) = v, (3) = w}))

Vector[column](%id = 36893490965311563044)

 (3)
> 

Now that p_ni has been transformed and p_n is the transformed point, m_m is the position vector of a point on the bottom plate. These points are not moving at all, and each one is described by a parameter mu_m
> 

m[m] := R*(Vector(3, {(1) = cos(`μ__m`), (2) = -sin(`μ__m`), (3) = h/R}))

Vector[column](%id = 36893490965244196612)

 (4)
> 

with(LinearAlgebra)

[`&x`, Add, Adjoint, BackwardSubstitute, BandMatrix, Basis, BezoutMatrix, BidiagonalForm, BilinearForm, CARE, CharacteristicMatrix, CharacteristicPolynomial, Column, ColumnDimension, ColumnOperation, ColumnSpace, CompanionMatrix, CompressedSparseForm, ConditionNumber, ConstantMatrix, ConstantVector, Copy, CreatePermutation, CrossProduct, DARE, DeleteColumn, DeleteRow, Determinant, Diagonal, DiagonalMatrix, Dimension, Dimensions, DotProduct, EigenConditionNumbers, Eigenvalues, Eigenvectors, Equal, ForwardSubstitute, FrobeniusForm, FromCompressedSparseForm, FromSplitForm, GaussianElimination, GenerateEquations, GenerateMatrix, Generic, GetResultDataType, GetResultShape, GivensRotationMatrix, GramSchmidt, HankelMatrix, HermiteForm, HermitianTranspose, HessenbergForm, HilbertMatrix, HouseholderMatrix, IdentityMatrix, IntersectionBasis, IsDefinite, IsOrthogonal, IsSimilar, IsUnitary, JordanBlockMatrix, JordanForm, KroneckerProduct, LA_Main, LUDecomposition, LeastSquares, LinearSolve, LyapunovSolve, Map, Map2, MatrixAdd, MatrixExponential, MatrixFunction, MatrixInverse, MatrixMatrixMultiply, MatrixNorm, MatrixPower, MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial, Minor, Modular, Multiply, NoUserValue, Norm, Normalize, NullSpace, OuterProductMatrix, Permanent, Pivot, PopovForm, ProjectionMatrix, QRDecomposition, RandomMatrix, RandomVector, Rank, RationalCanonicalForm, ReducedRowEchelonForm, Row, RowDimension, RowOperation, RowSpace, ScalarMatrix, ScalarMultiply, ScalarVector, SchurForm, SingularValues, SmithForm, SplitForm, StronglyConnectedBlocks, SubMatrix, SubVector, SumBasis, SylvesterMatrix, SylvesterSolve, ToeplitzMatrix, Trace, Transpose, TridiagonalForm, UnitVector, VandermondeMatrix, VectorAdd, VectorAngle, VectorMatrixMultiply, VectorNorm, VectorScalarMultiply, ZeroMatrix, ZeroVector, Zip]

 (5)

Here the distance between the points on the top plate and the bottom plate is calculated, and the distance is left as a function of all the problem's variables and each leg's parameters: ρ_n and μ_n.

> 

`Δp` := p__n-m[m]

Vector[column](%id = 36893490965243727012)

 (6)
> 

with(VectorCalculus)

[`&x`, `*`, `+`, `-`, `.`, `<,>`, `<|>`, About, AddCoordinates, ArcLength, BasisFormat, Binormal, ConvertVector, CrossProduct, Curl, Curvature, D, Del, DirectionalDiff, Divergence, DotProduct, Flux, GetCoordinateParameters, GetCoordinates, GetNames, GetPVDescription, GetRootPoint, GetSpace, Gradient, Hessian, IsPositionVector, IsRootedVector, IsVectorField, Jacobian, Laplacian, LineInt, MapToBasis, Nabla, Norm, Normalize, PathInt, PlotPositionVector, PlotVector, PositionVector, PrincipalNormal, RadiusOfCurvature, RootedVector, ScalarPotential, SetCoordinateParameters, SetCoordinates, SpaceCurve, SurfaceInt, TNBFrame, TangentLine, TangentPlane, TangentVector, Torsion, Vector, VectorField, VectorPotential, VectorSpace, Wronskian, diff, eval, evalVF, int, limit, series]

 (7)
> 

s[n] := sqrt(`&Delta;p`.`&Delta;p`)

((cos(theta)*cos(psi)*R*cos(rho[n])-cos(theta)*sin(psi)*R*sin(rho[n])+u-R*cos(mu__m))^2+((sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*cos(rho[n])-(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*sin(rho[n])+v+R*sin(mu__m))^2+((cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*cos(rho[n])-(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*sin(rho[n])+w-h)^2)^(1/2)

 (8)

Here the parameters for each leg are defined, and with these values, each leg is given it's distance.

> 

rho[1] := -(1/12)*Pi; rho[2] := (1/12)*Pi; rho[3] := 2*Pi*(1/3)-(1/12)*Pi; rho[4] := (1/12)*Pi+2*Pi*(1/3); rho[5] := 4*Pi*(1/3)-(1/12)*Pi; rho[6] := 4*Pi*(1/3)+(1/12)*Pi

-(1/12)*Pi

  

(1/12)*Pi

  

(7/12)*Pi

  

(3/4)*Pi

  

(5/4)*Pi

  

(17/12)*Pi

 (9)
> 

mu[1] := 0; mu[2] := 2*Pi*(1/3); mu[3] := 4*Pi*(1/3)

0

  

(2/3)*Pi

  

(4/3)*Pi

 (10)
> 

s[1] := eval(s[n], {`&mu;__m` = mu[1], rho[n] = rho[1]})

((cos(theta)*cos(psi)*R*cos((1/12)*Pi)+cos(theta)*sin(psi)*R*sin((1/12)*Pi)+u-R)^2+((sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*cos((1/12)*Pi)+(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*sin((1/12)*Pi)+v)^2+((cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*cos((1/12)*Pi)+(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*sin((1/12)*Pi)+w-h)^2)^(1/2)

 (11)
> 

s[2] := eval(s[n], {`&mu;__m` = mu[1], rho[n] = rho[2]})

((cos(theta)*cos(psi)*R*cos(rho[m])-cos(theta)*sin(psi)*R*sin(rho[m])+u-R*cos(mu__m))^2+((sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*cos(rho[m])-(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*sin(rho[m])+v+R*sin(mu__m))^2+((cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*cos(rho[m])-(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*sin(rho[m])+w-h)^2)^(1/2)

 (12)
> 

s[3] := eval(s[n], {`&mu;__m` = mu[2], rho[n] = rho[3]})

((-cos(theta)*cos(psi)*R*cos((5/12)*Pi)-cos(theta)*sin(psi)*R*sin((5/12)*Pi)+u+(1/2)*R)^2+(-(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*cos((5/12)*Pi)-(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*sin((5/12)*Pi)+v+(1/2)*R*3^(1/2))^2+(-(cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*cos((5/12)*Pi)-(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*sin((5/12)*Pi)+w-h)^2)^(1/2)

 (13)
> 

s[4] := eval(s[n], {`&mu;__m` = mu[2], rho[n] = rho[4]})

((-(1/2)*cos(theta)*cos(psi)*R*2^(1/2)-(1/2)*cos(theta)*sin(psi)*R*2^(1/2)+u+(1/2)*R)^2+(-(1/2)*(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*2^(1/2)-(1/2)*(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*2^(1/2)+v+(1/2)*R*3^(1/2))^2+(-(1/2)*(cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*2^(1/2)-(1/2)*(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*2^(1/2)+w-h)^2)^(1/2)

 (14)
> 

s[5] := eval(s[n], {`&mu;__m` = mu[3], rho[n] = rho[5]})

((-(1/2)*cos(theta)*cos(psi)*R*2^(1/2)+(1/2)*cos(theta)*sin(psi)*R*2^(1/2)+u+(1/2)*R)^2+(-(1/2)*(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*2^(1/2)+(1/2)*(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*2^(1/2)+v-(1/2)*R*3^(1/2))^2+(-(1/2)*(cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*2^(1/2)+(1/2)*(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*2^(1/2)+w-h)^2)^(1/2)

 (15)
> 

s[6] := eval(s[n], {`&mu;__m` = mu[3], rho[n] = rho[6]})

((-cos(theta)*cos(psi)*R*cos((5/12)*Pi)+cos(theta)*sin(psi)*R*sin((5/12)*Pi)+u+(1/2)*R)^2+(-(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*cos((5/12)*Pi)+(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*sin((5/12)*Pi)+v-(1/2)*R*3^(1/2))^2+(-(cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*cos((5/12)*Pi)+(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*sin((5/12)*Pi)+w-h)^2)^(1/2)

 (16)
> 

equations := [-d[1]^2+s[1]^2 = 0, -d[2]^2+s[2]^2 = 0, -d[3]^2+s[3]^2 = 0, -d[4]^2+s[4]^2 = 0, -d[5]^2+s[5]^2 = 0, -d[6]^2+s[6]^2 = 0]

[(cos(theta)*cos(psi)*R*cos((1/12)*Pi)+cos(theta)*sin(psi)*R*sin((1/12)*Pi)+u-R)^2+((sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*cos((1/12)*Pi)+(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*sin((1/12)*Pi)+v)^2+((cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*cos((1/12)*Pi)+(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*sin((1/12)*Pi)+w-h)^2-d[1]^2 = 0, -d[2]^2+s[2]^2 = 0, (-cos(theta)*cos(psi)*R*cos((5/12)*Pi)-cos(theta)*sin(psi)*R*sin((5/12)*Pi)+u+(1/2)*R)^2+(-(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*cos((5/12)*Pi)-(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*sin((5/12)*Pi)+v+(1/2)*R*3^(1/2))^2+(-(cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*cos((5/12)*Pi)-(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*sin((5/12)*Pi)+w-h)^2-d[3]^2 = 0, (-(1/2)*cos(theta)*cos(psi)*R*2^(1/2)-(1/2)*cos(theta)*sin(psi)*R*2^(1/2)+u+(1/2)*R)^2+(-(1/2)*(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*2^(1/2)-(1/2)*(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*2^(1/2)+v+(1/2)*R*3^(1/2))^2+(-(1/2)*(cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*2^(1/2)-(1/2)*(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*2^(1/2)+w-h)^2-d[4]^2 = 0, (-(1/2)*cos(theta)*cos(psi)*R*2^(1/2)+(1/2)*cos(theta)*sin(psi)*R*2^(1/2)+u+(1/2)*R)^2+(-(1/2)*(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*2^(1/2)+(1/2)*(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*2^(1/2)+v-(1/2)*R*3^(1/2))^2+(-(1/2)*(cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*2^(1/2)+(1/2)*(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*2^(1/2)+w-h)^2-d[5]^2 = 0, (-cos(theta)*cos(psi)*R*cos((5/12)*Pi)+cos(theta)*sin(psi)*R*sin((5/12)*Pi)+u+(1/2)*R)^2+(-(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))*R*cos((5/12)*Pi)+(sin(phi)*sin(theta)*sin(psi)+cos(phi)*cos(psi))*R*sin((5/12)*Pi)+v-(1/2)*R*3^(1/2))^2+(-(cos(phi)*sin(theta)*cos(psi)+sin(psi)*sin(phi))*R*cos((5/12)*Pi)+(cos(phi)*sin(theta)*sin(psi)-sin(phi)*cos(psi))*R*sin((5/12)*Pi)+w-h)^2-d[6]^2 = 0]

 (17)
> 

variables := [u, v, w, psi, theta, phi]

[u, v, w, psi, theta, phi]

 (18)
> 

parameters := [d[1], d[2], d[3], d[4], d[5], d[6]]

[d[1], d[2], d[3], d[4], d[5], d[6]]

 (19)
> 

solve(equations, variables)

[]

 (20)
 

NULL

Download FiguringThingsOut.mw

@C_R, first of all, excuse me for not wr...

Ohmyus 15
June 25 2024
0 0

@C_R, first of all, excuse me for not writing things down clearly. I'm not sure wether my application is the forward or inverse kinematics. 

I do want to simulate a Stewart platform, but instead of calculating the length of each leg given the desired position of the platform, I want to calculate the postion and rotation of the platform given the length of each leg. If that is the forward kinematics problem then I got it wrong on the original question and will correct it. Here's an image of the problem, so it can be understood more easily. Also, I will add the Maple worksheet to the original post.

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