Consider the frames {0}, {A} and {B} placed on a rectangular parallelepiped
as shown in Figure 1. Axes XB
and YB
intersect the axis Y0
. Determine
find the answer no need to explain MATLAB code, remember this problem is from ROBOTICS: MECHANICS AND CONTROL from kinematics part and it will teach to b.tech mechanical final year student---**Question Stem:**
Q. 5. Consider the frames {0}, {A} and {B} placed on a rectangular parallelepiped as shown in Figure 1. Axes X_B and Y_B intersect the axis Y_0. Determine $_0R$, $_B^0R$ and $_B^AR$. Also compute the transformations and draw the corresponding Figures using MATLAB.
**Chart/Diagram Description:**
* **Type:** 3D geometric diagram showing a rectangular parallelepiped with three attached coordinate frames.
* **Main Elements:**
* **Rectangular Parallelepiped:** A 3D box.
* Dimensions are indicated: Length 3 units along the X0 axis, width 2 units along the Y0 axis, and height 1 unit along the Z0 axis.
* **Coordinate Frame {0}:**
* Origin: Labeled 'O', positioned at the bottom-left-front corner of the parallelepiped.
* Axes: Solid black lines with arrows.
* X0-axis: Extends horizontally to the right.
* Y0-axis: Extends diagonally upwards and to the left.
* Z0-axis: Extends vertically upwards, perpendicular to the X0-Y0 plane.
* Label: '{0}' is placed near the origin O.
* **Coordinate Frame {A}:**
* Origin: Labeled 'A', positioned on the bottom face of the parallelepiped, at the back-right corner (relative to frame {0}'s origin).
* Axes: Dashed red lines with arrows.
* XA-axis: Extends diagonally downwards and to the right, appearing parallel to the negative Y0 direction.
* YA-axis: Extends horizontally to the left, appearing parallel to the negative X0 direction.
* Label: '{A}' is placed near the origin A.
* **Coordinate Frame {B}:**
* Origin: Labeled 'B', positioned at the top-front-right corner of the parallelepiped.
* Axes: Dashed blue lines with arrows.
* XB-axis: Extends diagonally upwards and to the left, appearing parallel to the Y0 axis.
* YB-axis: Extends horizontally to the right, appearing parallel to the X0 axis.
* Label: '{B}' is placed near the origin B.
* **Labels and Annotations:**
* Question number: "Q. 5."
* Figure caption: "Figure 1."
* Points: O, A, B.
* Axes: X0, Y0, Z0, XA, YA, XB, YB.
* Frame labels: {0}, {A}, {B}.
* Dimensions: Numeric labels '3', '2', '1' indicating the lengths of the parallelepiped sides.
* **Relative Position and Direction:**
* The origin O of frame {0} is at the implied (0,0,0) of the system.
* The origin A of frame {A} is at the corner (3, 2, 0) relative to frame {0} (assuming dimensions are along X0, Y0, Z0 respectively). XA is visually parallel to -Y0, and YA is visually parallel to -X0.
* The origin B of frame {B} is at the corner (3, 0, 1) relative to frame {0}. XB is visually parallel to Y0, and YB is visually parallel to X0.
* The problem statement notes: "Axes X_B and Y_B intersect the axis Y_0." However, based on the visual representation, XB appears parallel to Y0, and YB appears parallel to X0, which would not intersect Y0 unless collinear or originating from the Y0 axis, respectively.
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答案文本
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In robotics kinematics, coordinate frames describe position and orientation in 3D space. We have a rectangular parallelepiped with three coordinate frames: frame zero at origin O, frame A at position A, and frame B at position B. Our task is to find the rotation matrices that describe the orientation relationships between these frames.
Frame zero is our reference coordinate frame located at origin O. It has standard orthogonal axes: X zero along the positive x direction, Y zero along positive y, and Z zero along positive z. The rotation matrix from frame zero to itself is simply the three by three identity matrix, representing no rotation.
Frame A is located at position A with coordinates (3,2,0). From the diagram, we observe that X_A is parallel to the negative Y₀ direction, Y_A is parallel to the negative X₀ direction, and Z_A remains parallel to Z₀. This configuration represents a 180-degree rotation about the Z-axis, giving us the rotation matrix zero R A.
Frame B is located at position B with coordinates (3,0,1). The X_B axis is parallel to Y₀, Y_B is parallel to X₀, and Z_B remains parallel to Z₀. This represents a 90-degree rotation about the Z-axis. We can compute all required rotation matrices including the transformation from frame A to frame B by multiplying the appropriate matrices. These rotation matrices are fundamental for robotic kinematics and control applications.