Problem 2 In terms of the xs, ŷs, 2s coordinates of a fixed space frame {s}, frame {a} has its x-axis pointing in the direction (0, 0, 1) and its y-axis pointing in the direction (-1,0,0), and frame {b} has its x-axis pointing in the direction (1,0,0) and its ŷ₁-axis pointing in the direction (0,0,−1). The origin of {a} is at (3, 0, 0) in {s} and the origin of {b} is at (0, 2, 0) in {s}. (a) Draw by hand a diagram showing {a} and {b} relative to {s}. (b) Write down the rotation matrices Rsa and Rsb and the transformation matrices Tsa and Tsb.

College Physics
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Chapter1: Units, Trigonometry. And Vectors
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**Problem 2: Coordinate Frame Analysis**

In terms of the \( \hat{x}_s, \hat{y}_s, \hat{z}_s \) coordinates of a fixed space frame \(\{s\}\), frame \(\{a\}\) has its \( \hat{x}_a\)-axis pointing in the direction \((0, 0, 1)\) and its \( \hat{y}_a\)-axis pointing in the direction \((-1, 0, 0)\). Frame \(\{b\}\) has its \( \hat{x}_b\)-axis pointing in the direction \((1, 0, 0)\) and its \( \hat{y}_b\)-axis pointing in the direction \((0, 0, -1)\). The origin of \(\{a\}\) is at \((3, 0, 0)\) in \(\{s\}\) and the origin of \(\{b\}\) is at \((0, 2, 0)\) in \(\{s\}\).

**Tasks:**

(a) *Draw by hand a diagram showing \(\{a\}\) and \(\{b\}\) relative to \(\{s\}\).*

(b) *Write down the rotation matrices \(R_{sa}\) and \(R_{sb}\) and the transformation matrices \(T_{sa}\) and \(T_{sb}\).*

(c) *Calculate the matrix exponential corresponding to the exponential coordinates of rigid-body motion \(S \theta = (0, 1, 2, 3, 0, 0)\). Draw the corresponding frame relative to \(\{s\}\), as well as the screw axis \(S\).*
Transcribed Image Text:**Problem 2: Coordinate Frame Analysis** In terms of the \( \hat{x}_s, \hat{y}_s, \hat{z}_s \) coordinates of a fixed space frame \(\{s\}\), frame \(\{a\}\) has its \( \hat{x}_a\)-axis pointing in the direction \((0, 0, 1)\) and its \( \hat{y}_a\)-axis pointing in the direction \((-1, 0, 0)\). Frame \(\{b\}\) has its \( \hat{x}_b\)-axis pointing in the direction \((1, 0, 0)\) and its \( \hat{y}_b\)-axis pointing in the direction \((0, 0, -1)\). The origin of \(\{a\}\) is at \((3, 0, 0)\) in \(\{s\}\) and the origin of \(\{b\}\) is at \((0, 2, 0)\) in \(\{s\}\). **Tasks:** (a) *Draw by hand a diagram showing \(\{a\}\) and \(\{b\}\) relative to \(\{s\}\).* (b) *Write down the rotation matrices \(R_{sa}\) and \(R_{sb}\) and the transformation matrices \(T_{sa}\) and \(T_{sb}\).* (c) *Calculate the matrix exponential corresponding to the exponential coordinates of rigid-body motion \(S \theta = (0, 1, 2, 3, 0, 0)\). Draw the corresponding frame relative to \(\{s\}\), as well as the screw axis \(S\).*
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