The roll, pitch and yaw of an object relative to another is complex to compute. We use 3D complex number to compute them which makes the computation easier and more intuitive.<\/p>\n\n\n\n
Roll, pitch and yaw are angles of orientation of an object in\nspace and the conversion of these angles among different reference frames is\nnot easy, as this example illustrates:<\/p>\n\n\n\n
https:\/\/math.stackexchange.com\/questions\/1884215\/how-to-calculate-relative-pitch-roll-and-yaw-given-absolutes<\/a><\/p>\n\n\n\n We will use 3D complex number to compute the roll,\npitch and yaw of a telephon relative to a car. We label the car as object a and the telephon as object\nb. Their reference frames are labeled as\nframe A and B and these frames are orientated\nwith respect to the ground whose frame is\nlabeled as frame G. The base vectors of the frame A are a<\/strong>x<\/sub>, a<\/strong>y<\/sub> and a<\/strong>z<\/sub>,\nthat of the frame Bare b<\/strong>x<\/sub>, b<\/strong>y<\/sub> and b<\/strong>z<\/sub> and that of the\nframe G are g<\/strong>x<\/sub>, g<\/strong>y<\/sub>\nand g<\/strong>z<\/sub>. The vectors a<\/strong>x<\/sub>, a<\/strong>y<\/sub> and a<\/strong>z<\/sub>\nand b<\/strong>x<\/sub>, b<\/strong>y<\/sub> and b<\/strong>z<\/sub> are expressedwith g<\/strong>x<\/sub>, g<\/strong>y<\/sub>\nand g<\/strong>z<\/sub> in equations (1) and (2) with M<\/strong>a<\/sub>and M<\/strong>b<\/sub> being the matrices of transformation.<\/p>\n\n\n\n The roll, pitch and yaw of the\nobject b relative to the object a define the\norientation of the frame B in the frame A.<\/strong> This orientation is defined by the angles of rotation q, j and y, see Figure 1. For computing these angles we express the vectors b<\/strong>x<\/sub>, b<\/strong>y<\/sub> and b<\/strong>z<\/sub>\nwith the vectors a<\/strong>x<\/sub>, a<\/strong>y<\/sub> and a<\/strong>z<\/sub> in (3), (4) and (5). In (10), (11) and (12) the coefficients in the parentheses are dot products between\na<\/strong>x<\/sub>, a<\/strong>y<\/sub> and a<\/strong>z<\/sub>\nand b<\/strong>x<\/sub>, b<\/strong>y<\/sub> and b<\/strong>z<\/sub>. The equations (10) , (11) and (12) are written in matrix form in\n(13) from which we extract the\nmatrix of transformation M<\/strong>ba<\/sub> shown in (14), see (3), (4) and (5). <\/p>\n\n\n\n From the 5 underscored coefficients in (14) we\nderive the angles q, j and y using (15), (16) and (17), knowing that \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n is always\npositive because j is between -p\/2 and p\/2, see Figure 1. Roll, pitch and yaw of an object are the rotation\nangles of the object around the x, y and z axis respectively, see Figure 1 and the page https:\/\/en.wikipedia.org\/wiki\/Flight_dynamics_(fixed-wing_aircraft)<\/a>. <\/p>\n\n\n\n So, the conversion formula between roll, pitch and yaw and the angles q, j and y is equation (18). <\/p>\n\n\n\n The main advancement given by our method using 3D complex number is to have related the roll, pitch and yaw of an\nobject to the 3D complex number that represent the\nx axis <\/strong>of the object. This mathematical discovery makes the\nconversion of roll, pitch and yaw between\narbitrary frames easier and more intuitive.<\/p>\n\n\n\n The angles of\nrotation q, j and y are computed with the 5 underscored\ncoefficients of the matrix M<\/strong>ba<\/sub> in (14) saving thus the computation time for the\n4 other coefficients which are not needed.\nThis will make the computation faster and the motion of moving\nimages on screen smoother. So, our method using 3D complex number would be\nbeneficial to applications such as video\ngames or street view of digital map etc.<\/p>\n\n\n\n For more detail\nof this\nstudy please read the complete paper here:<\/p>\n\n\n\n \u00abDetermination\nof the relative roll, pitch and yaw<\/a> between\narbitrary objects using 3D\ncomplex number<\/a>\u00bb<\/p>\n\n\n\n https:\/\/www.academia.edu\/92242546\/Determination_of_the_relative_roll_pitch_and_yaw_between_arbitrary_objects_using_3D_complex_number<\/a> \n\nKuan Peng\n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":" The roll, pitch and yaw of an object relative to another is complex to compute. We use 3D complex number to compute them which makes the computation easier and more intuitive. Roll, pitch and yaw are angles of orientation of … Continue reading