1. Consider the schematic shown in the figure below, representing a 2-piece robotic arm. Find a formula for the vector r from the origin to the endpoint of the arm in terms of the radii R₁, R2 and the angles 01, 02. Use this formula to show that, if 0₁ = 02, then the set of all possible positions of the point P forms an ellipse. R₂ 0₁ 02 P R₁ 2. Suppose the vector 7= (2, 1) and = (1/2,2). Refer to the figure on the right and find the following: i. ||v ū ii. Lu V iii. *: the reflection of u across u
1. Consider the schematic shown in the figure below, representing a 2-piece robotic arm. Find a formula for the vector r from the origin to the endpoint of the arm in terms of the radii R₁, R2 and the angles 01, 02. Use this formula to show that, if 0₁ = 02, then the set of all possible positions of the point P forms an ellipse. R₂ 0₁ 02 P R₁ 2. Suppose the vector 7= (2, 1) and = (1/2,2). Refer to the figure on the right and find the following: i. ||v ū ii. Lu V iii. *: the reflection of u across u
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Educational Website: Advanced Robotics and Vector Analysis
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**1. Robotic Arm Position Analysis**
Consider the schematic shown in the figure below, representing a 2-piece robotic arm. Find a formula for the vector \( \mathbf{r} \) from the origin to the endpoint of the arm in terms of the radii \( R_1 \), \( R_2 \) and the angles \( \theta_1 \), \( \theta_2 \).
**Formula Derivation:**
\[ \mathbf{r} = R_1 \cos(\theta_1) + R_2 \cos(\theta_2), \, R_1 \sin(\theta_1) + R_2 \sin(\theta_2) \]
Use this formula to show that, if \( \theta_1 = \theta_2 \), then the set of all possible positions of the point \( P \) forms an ellipse.
**Graph Explanation:**
The diagram illustrates the 2-piece robotic arm with lengths \( R_1 \) and \( R_2 \). Point \( P \) represents the endpoint of the robotic arm. The angles \( \theta_1 \) and \( \theta_2 \) define the positions of each segment of the arm, creating a vector \( \mathbf{r} \) from the origin to point \( P \).
---
**2. Vector Projections and Reflections**
Suppose the vector \( \mathbf{v} = \langle 2, 1 \rangle \) and \( \mathbf{u} = \langle 1/2, 2 \rangle \). Refer to the figure on the right and find the following:
**i. \( \mathbf{u}_{\parallel \mathbf{v}} \)**
**ii. \( \mathbf{u}_{\perp \mathbf{v}} \)**
**iii. \( \mathbf{u}^* \): the reflection of \( \mathbf{u} \) across \( \mathbf{v} \)**
**Graph Explanation:**
The diagram shows the vectors \( \mathbf{u} \) and \( \mathbf{v} \) along with their projections and the reflection of \( \mathbf{u} \). The right-angle coordinates and reflection lines illustrate the geometric relationships between the vectors.
---
In summary, this lesson involves deriving vector formulas for a 2-piece robotic arm,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0761ca0b-8a30-47ed-b77e-00687bf0302e%2Faa8f96b5-95af-45b5-a1cb-b7c206930527%2F06ofyxi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Website: Advanced Robotics and Vector Analysis
---
**1. Robotic Arm Position Analysis**
Consider the schematic shown in the figure below, representing a 2-piece robotic arm. Find a formula for the vector \( \mathbf{r} \) from the origin to the endpoint of the arm in terms of the radii \( R_1 \), \( R_2 \) and the angles \( \theta_1 \), \( \theta_2 \).
**Formula Derivation:**
\[ \mathbf{r} = R_1 \cos(\theta_1) + R_2 \cos(\theta_2), \, R_1 \sin(\theta_1) + R_2 \sin(\theta_2) \]
Use this formula to show that, if \( \theta_1 = \theta_2 \), then the set of all possible positions of the point \( P \) forms an ellipse.
**Graph Explanation:**
The diagram illustrates the 2-piece robotic arm with lengths \( R_1 \) and \( R_2 \). Point \( P \) represents the endpoint of the robotic arm. The angles \( \theta_1 \) and \( \theta_2 \) define the positions of each segment of the arm, creating a vector \( \mathbf{r} \) from the origin to point \( P \).
---
**2. Vector Projections and Reflections**
Suppose the vector \( \mathbf{v} = \langle 2, 1 \rangle \) and \( \mathbf{u} = \langle 1/2, 2 \rangle \). Refer to the figure on the right and find the following:
**i. \( \mathbf{u}_{\parallel \mathbf{v}} \)**
**ii. \( \mathbf{u}_{\perp \mathbf{v}} \)**
**iii. \( \mathbf{u}^* \): the reflection of \( \mathbf{u} \) across \( \mathbf{v} \)**
**Graph Explanation:**
The diagram shows the vectors \( \mathbf{u} \) and \( \mathbf{v} \) along with their projections and the reflection of \( \mathbf{u} \). The right-angle coordinates and reflection lines illustrate the geometric relationships between the vectors.
---
In summary, this lesson involves deriving vector formulas for a 2-piece robotic arm,
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