2. 3D Rotations of a Rigid Body About XYZaxes
Referring to the original convention of Figure 1 assume that the length of the axis is equal to one unit, X (x,y,z) or Y (x,y,z) or Z (x,y,z) is fixed while A (a,b,c) or B (a,b,c) or C (a,b,c) is mobile. The counterclockwise rotation is taken to be positive.
2.1. Rotation about Xaxis
After rotating B,
After rotating C,
Before  After 

\(A(a,b,c) = (1,0,0)\) 
\(A(a,b,c) = (1,0,0)\) 
\(B(a,b,c) = (0,1,0)\) 
\(B(a,b,c) = (0,B \cos(\alpha),B \sin(\alpha))\) 
\(C(a,b,c) = (0,0,1)\) 
\(C(a,b,c) = (0,C \sin(\alpha), C \cos(\alpha))\) 
A  B  C 

\(+A\) 
\(0\) 
\(0\) 
\(0\) 
\(+B \cos(\alpha)\) 
\(C \sin(\alpha)\) 
\(0\) 
\(+B \sin(\alpha)\) 
\(+C \cos(\alpha)\) 
\((1,0,0)\) 
\((0,B \cos(\alpha),B \sin(\alpha))\) 
\((0,C \sin(\alpha), C \cos(\alpha))\) 
The rotation matrix for a rotation about the xaxis is given by:
\[\begin{align}
R(x, \alpha) = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos(\alpha) & \sin(\alpha) \\
0 & \sin(\alpha) & \cos(\alpha)
\end{array}\right]
\end{align}\]

2.2. Rotation about Yaxis
After rotating A,
After rotating C,
Before  After 

\(A(a,b,c) = (1,0,0)\) 
\(A(a,b,c) = (A \cos(\beta),0,A sin(\beta))\) 
\(B(a,b,c) = (0,1,0)\) 
\(B(a,b,c) = (0,1,0)\) 
\(C(a,b,c) = (0,0,1)\) 
\(C(a,b,c) = (C \sin(\beta),0, C \cos(\beta))\) 
A  B  C 

\(+A \cos(\beta)\) 
\(0\) 
\(+C \sin(\beta)\) 
\(0\) 
\(+B\) 
\(0\) 
\(A \sin(\beta)\) 
\(0\) 
\(+C \cos(\beta)\) 
\((+A \cos(\beta),0,A \sin(\beta))\) 
\((0,+B,0)\) 
\((+C \sin(\beta),0, +C \cos(\beta))\) 
The rotation matrix for a rotation about the yaxis is given by:
\[\begin{align}
R(y, \beta) = \left[\begin{array}{ccc}
\cos(\beta) & 0 & \sin(\beta) \\
0 & 1 & 0 \\
\sin(\beta) & 0 & \cos(\beta)
\end{array}\right]
\end{align}\]

2.3. Rotation about Zaxis
After rotating A,
After rotating B,
Before  After 

\(A(a,b,c) = (1,0,0)\) 
\(A(a,b,c) = (A \cos(\gamma),A sin(\gamma),0)\) 
\(B(a,b,c) = (0,1,0)\) 
\(B(a,b,c) = (B \sin(\gamma),B \cos(\gamma),0)\) 
\(C(a,b,c) = (0,0,1)\) 
\(C(a,b,c) = (0,0,1)\) 
A  B  C 

\(+A \cos(\gamma)\) 
\(B \sin(\gamma)\) 
\(0\) 
\(A \sin(\gamma)\) 
\(+B \cos(\gamma)\) 
\(0\) 
\(0\) 
\(0\) 
\(1\) 
\((+A \cos(\gamma),A \sin(\gamma),0)\) 
\((B \sin(\gamma),+B \cos(\gamma),0)\) 
\((0,0, 1)\) 
The rotation matrix for a rotation about the zaxis is given by:
\[\begin{align}
R(z, \gamma) = \left[\begin{array}{ccc}
\cos(\gamma) & \sin(\gamma) & 0 \\
\sin(\gamma) & \cos(\gamma) & 0 \\
0 & 0 & 1
\end{array}\right]
\end{align}\]

where \(\alpha, \beta, \gamma\) is any angle of rotation.
These formulas can be used to calculate the rotation matrix for any rotation about the x, y, or zaxis in 3D space. The angle of rotation is specified in radians.
2.4. Coordinates of a Point Rotated About an Axis
Example 1: Rotation about Xaxis 

The coordinates of a point \(q_{abc} = (3, 7, 5)^T\) and is rotated about the OXaxis of the reference frame OXYZ, by an angle of \(60^o\). Determine the coordinates of the point \(q_{xyz}\) 
Solution:
\[q_{xyz} = R(x,60^o)q_{abc}\]

Example 2: Rotation about Yaxis 

The coordinate of a point \(p_{abc}\) in the mobile frame OABC is given by \((4, 4, 2\sqrt{3})^T\). If the frame OABC is rotated by \(60^o\) with respect to OY of the OXYZ frame, find the coordinate of \(p_{xyz}\) with respect to the base frame. 
Solution:
\[p_{xyz} = R(y,60^o)p_{abc}\]

Example 3: Rotation about Zaxis 

The coordinate of a point \(p_{abc} = (7, 6, 5)^T\) in the body coordinate frame OABC is rotated \(30^o\) about OZaxis. Determine the coordinates of the vector \(p_{xyz}\) with respect to the base reference coordinate frame. 
Solution:
\[p_{xyz} = R(x,60^o)p_{abc}\]

2.4.1. Axis and Angle of Rotation
Example 4 

For the following rotation matrix, determine the axis of rotation and the angle of the rotation about the same. 
Solution:
Please attempt. 
2.5. Coordinates of a Point Relative to a Mobile Frame
Example 5 

A mobile body reference frame OABC is rotated \(60^o\) about OYaxis of the fixed base reference frame OXYZ. If \(p_{xyz} = (2, 3, 6)^T\) and \(q_{xyz} = (4, 2, 5)^T\) are the coordinates with respect to OXYZ plane, what are the corresponding coordinates of \(p\) and \(q\) with respect to OABC frame? 
Solution:
\[p_{abc} = R(y,60^o)^{1}p_{xyz} = R(y,60^o)^{T}p_{xyz} \\
p_{abc} = R(y,60^o)^{1}p_{xyz} = R(y,60^o)^{T}p_{xyz}\]

Example 6 

The coordinates of point \(Q\) with respect to the base reference frame are given by \((4, 2\sqrt{3}, 5)^T\). Determine the coordinates of \(Q\) with respect to the mobile rotated frame of the robot if the angle of rotation with the OX is \(60^o\). 
Solution:
\[Q_{abc} = R(x,60^o)^{1}Q_{xyz} = R(x,60^o)^{T}Q_{xyz}\]

Example 7: A singleaxis robot 

Take a singleaxis robot with a fixed base and a mobile link. Suppose the mobile frame has a point \(p_M\) given by \((2, 2, 8)^T\). Find the coordinates of the point \(p_F\) with respect to the base frame when \(\theta_1 = 180^o\) and \(\theta_2 = 0^o\) 
Solution:
\[(p_{F})_{\theta = 180^o} = R(z,180^o)p_{M} \\
(p_{F})_{\theta = 0^o} = R(z,0^o)p_{M}\]

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Comments (4)
Using the knowledge of spatial mechanics and computer programming introduced in this course, you can go on to perform advanced kinematic analysis of 3D mechanisms.
These skills apply to robotics systems design, research, and development.
The notes are on point. I've gained alot of skill and visualization from the unit.
Does it mean the inverse of a 3x3 matrix same as its transpose
In spatial mechanics, the inverse and transpose of a 3x3 matrix are not the same, but they can be related in specific cases. The inverse (A⁻¹) is a matrix that, when multiplied by A, results in the identity matrix (A * A⁻¹ = I). The transpose (Aᵀ) is obtained by interchanging the rows and columns of A.
For 3x3 orthogonal matrices, often used to represent rotations, the transpose is equal to the inverse (A⁻¹ = Aᵀ) because they satisfy Aᵀ * A = I. This property is unique to orthogonal matrices, and for general 3x3 matrices, the inverse and transpose are not the same.