Matlab matricies

Rotations is significant, for example Rotate 90 degrees about x axis Rotate 90 degrees about y axis Rotate -90 degrees about x axisThis gives 90 degree rotation about Z axis,whereas Rotate 90 degrees about x axis Rotate -90 degrees about x axis Rotate 90 degrees about y axisThis gives 90 degree rotation about y axis (first 2 lines cancel out).Because this is so difficult, it is usual to convert to matrix notation or quaternions and calculate the product and then convert back to euler angles. But if this is done many times the rounding errors of all these conversions will build up, leading to distortions. So if we need to do a lot of such calculations it may be best to work entirely in matricies or quaternions even though they are less intuitive than Euler angles.I can think of different ways to combine rotations so I have put a discussion of this here.Rotation about single axisFirst we will consider rotation purely about each of the three axes individually, then we will consider the different ways of combining them.About Z axisFirst assume a rotation purely around the z axis, measuring from the x axis, as shown here:In order to combine rotations using matricies we need to be clear about what conventions that we are using. The coordinate directions are represented by a right hand coordinate system and the rotation directions are represented by a the right hand rule . The identification of cells of matrix and ordering of rows and columns (as explained here) is as follows: = m00 m01 m02 m10 m11 m12 m20 m21 m22 So point x=1, y=0, z=0is transformed to x=cos(heading), y =sin(heading) , z=0and point x=0, y=1, z=0is transformed to x=-sin(heading), y =cos(heading) , z=0so the complete matrix for rotation about the z axis is: [R1] = cos(heading) = c φ -sin(heading) = -sφ 0 sin(heading) = sφ cos(heading) = c φ 0 0 0 1 About z axis using axis angleAxis = (0,0,1) Angle =φAbout z axis using quaternionsThe axis-angle above can be converted to quaternion as described here.cos(φ/2) + (0 i + 0 j + 1k) * sin(φ/2)cos(φ/2) + k * sin(φ/2)About Y axisSimilarly rotation about the y axis, measuring from the z axis, gives:So point x=1, y=0, z=0is transformed to x=cos(heading), y =0, z=- sin(heading)and point x=0, y=0, z=1is transformed to x=sin(heading), y=0, z =cos(heading) so the complete matrix for rotation about the z axis is: [R2] = cos(attitude) = c θ 0 sin(attitude) = sθ 0 1 0 -sin(attitude) = -sθ 0 cos(attitude) = c θ About y axis using axis angleAxis = (0,1,0) Angle =θAbout y axis using quaternionsThe axis-angle above can be converted to quaternion as described here.cos(θ/2) + (0

Shown here.NASA Standard Aeroplane (reversed order) adapted from a diagram from Andy angles: φ heading θ attitude ψ bank Coordinate System: right hand Order: x,y,z = [R1][R2][R3] This gives a combined transformational matrix of,[R] = [R3][R2][R1] [R] = c θ 0 s θ 0 1 0 -s θ 0 c θ multiplying matricies gives: [R] = c ψ*c θ sψ*cφ + c ψ*s θ*s φ sψ*sφ - c ψ* s θ*c φ -s ψ*c θ cψ*cφ -s ψ*s θ*s φ cψ*sφ + s ψ* s θ*c φ s θ -c θ *s φ c θ *c φ related pages: matrix euler to matrix conversion matrix to euler conversionStandard Aeroplane (reversed) using quaternionsWe can multiply the quaternions in order, as we did with the matricies:(cos(ψ/2) + k * sin(ψ/2)) * (cos(θ/2) + j * sin(θ/2)) * (cos(φ/2) + i * sin(φ/2))related pages: quaternions euler to quaternion conversion quaternion to euler conversionNASA Standard Aerospace angles: φ precession θ nutation ψ spin Coordinate System: right hand Order: z,y,z = [R3][R2][R1] In this case there is no individual rotation around the x axis, but the combination of rotation about the z axis and a rotation about the y axis can produce a rotation about the x axis, so a rotation about z then y then z can produce any possible rotation. [R1] = cos(precession) -sin(precession) 0 sin(precession) cos(precession) 0 0 0 1 [R2] = cos(nutation) 0 -sin(nutation) 0 1 0 sin(nutation) 0 cos(nutation) [R3] = cos(spin) -sin(spin) 0 sin(spin) cos(spin) 0 0 0 1 This gives a combined transformational matrix of,[R] = [R3][R2][R1]This is expanded out here. To save space cos(precession) is written as c ψ and so on: [R] = c θ 0 s θ 0 1 0 -s θ 0 c θ first multiply second two terms (for matrix multiplication see here) Remember order of matrix multiplication is significant. [R] = c θ *c φ -c θ *s φ s θ sφ cφ 0 -s θ*c φ s θ*s φ c θ related pages: matrix euler to matrix conversion matrix to euler conversionThe singularity is at:The Quaternion is: + i () + j () + k ()related pages: quaternions euler to quaternion conversion quaternion to euler conversionExampleIt is not always apparent that the three angles to specify a rotation are not independent of each other and must be applied in a certain order. For example imagine that we are aiming a dish at a satellite. The azimuth and elevation are independent of each other, for example we can aim south and then elevate up by the required inclination, or we can set the elevation and then turn and point toward the south. However there is a third angle, we can rotate about

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