WEBVTT

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To express the orientation of an object in the three dimensional space.

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So far we have used rotation matrices.

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With this tool, we are able to simplify and decompose the problem of orienting an object in the three

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dimensional space around three axes with three elementary rotations.

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With this tool, we are able to simplify and decompose the problem of orienting an object in the three

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dimensional space around three axes with three elementary rotations.

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A body can rotate around the Z axis.

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So if we call this rotation angle of the robot around the Z axis theta, then the orientation of the

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robot is defined by an elementary rotation matrix that we have calculated, and that is uniquely a function

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of the angle theta.

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Similarly, it can also rotate around the y axis this time, the angle that express the rotation of

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the object around the Y axis is called Phi.

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And once again, we can write an elementary rotation matrix that express the orientation of the robot

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with the respect to the y axis only.

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And finally, an object can also rotate around its x axis by an angle PC.

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The orientation of a robot around this axis can also be expressed with a new elementary rotation matrix

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that is only a function of the angle PC.

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If we consider the general case of a robot or generally of any object that can freely rotate and orient

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itself in the three dimensional space, then the representation of the orientation in Euler angles tells

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us that we can decompose the orientation of an object into three elementary components.

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Three elementary rotations around the X, Y, and Z axis.

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If we call theta the rotation angle around the Z axis Phi, the rotation angle around the Y axis and

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C the one around the X axis.

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For each of these elementary rotations, we know how to write the respective rotation matrix and therefore

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the overall orientation of an object is given by the composition.

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And so the product of these three elementary rotation matrices, these three elementary components of

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the orientation of an object in space are commonly known as roll pitch.

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And you.

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By multiplying these three matrices, we obtain a new matrix whose components will depend on the three

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angles.

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So Theta Phi and PSI.

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In conclusion, with this convention of Euler angles to express the rotation of an object in the three

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dimensional space, we need nine components.

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That is, to express the orientation of an object with rotation matrices.

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We need to calculate and store the nine components that form the rotation matrix.

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Furthermore, if we want to compose successive rotations, that is, if we want to multiply rotation

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matrices, we will have to perform a matrix product between two matrices with three rows and three columns.

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And this mean calculating 27 products and 18 sums.