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2 Problem Statement Consider a simplified two-joint robotic arm where the two joints are connected by rigid links. Each joint has a mass, and the links are modeled as idealized rods. The motion of each joint affects the other through the coupling of forces. The system can be modeled by the following second-order coupled ODES: m₁Ö₁ + k₁(01 — Orest) + k3 (01-02) = 0 (1) m202 + k2(02 - Orest) + k3 (02 - 01) = 0 where 01(t) and 02(t) represent the angular positions of the first and second joints, respec- tively. m₁ and m2 are the effective masses at each joint. k₁, k2 and k3 are the stiffness constants representing the elastic restoring forces in the joints and links. Orest 0 represents the equilibrium position for each joint. 3 Project Tasks: = The parameters are: m₁ = 1, m2 = 1, Orest = 0, k₁ = 1, k2 = 2, k3 = 3, Orest = 0 The initial conditions are: 01(0) = 1,02(0) = 2 A written report that includes: 1. A description of the solution method. 2. Plots of the joint positions over time. 3. An analysis of the oscillatory behavior for another different initial condition that you like.

2 Problem Statement Consider a simplified two-joint robotic arm where the two joints are connected by rigid links. Each joint has a mass, and the links are modeled as idealized rods. The motion of each joint affects the other through the coupling of forces. The system can be modeled by the following second-order coupled ODES: m₁Ö₁ + k₁(01 — Orest) + k3 (01-02) = 0 (1) m202 + k2(02 - Orest) + k3 (02 - 01) = 0 where 01(t) and 02(t) represent the angular positions of the first and second joints, respec- tively. m₁ and m2 are the effective masses at each joint. k₁, k2 and k3 are the stiffness constants representing the elastic restoring forces in the joints and links. Orest 0 represents the equilibrium position for each joint. 3 Project Tasks: = The parameters are: m₁ = 1, m2 = 1, Orest = 0, k₁ = 1, k2 = 2, k3 = 3, Orest = 0 The initial conditions are: 01(0) = 1,02(0) = 2 A written report that includes: 1. A description of the solution method. 2. Plots of the joint positions over time. 3. An analysis of the oscillatory behavior for another different initial condition that you like.

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
8th Edition
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Kreith, Frank; Manglik, Raj M.
Chapter4: Numerical Analysis Of Heat Conduction
Section: Chapter Questions
Problem 4.6P
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Consider a simplified two-joint robotic arm where the two joints are connected by rigid links.
Each joint has a mass, and the links are modeled as idealized rods. The motion of each joint
affects the other through the coupling of forces. The system can be modeled by the following
second-order coupled ODEs:
 
m1θ1" + k1(θ1 - θrest) + k3(θ1 - θ2) = 0 (equation 1)
m2θ2" + k2(θ2 - θrest) + k3(θ2 - θ1) = 0 (equation 2)
 
where θ1(t) and θ2(t) represent the angular positions of the first and second joints, respec-
tively. m1 and m2 are the effective masses at each joint. k1, k2 and k3 are the stiffness
constants representing the elastic restoring forces in the joints and links. θrest = 0 represents
the equilibrium position for each joint.
 
The parameters are:
m1 = 1, m2 = 1, θrest = 0, k1 = 1, k2 = 2, k3 = 3, θrest = 0
The initial conditions are:
θ1(0) = 1, θ2(0) = 2
 
Please solve by hand using Laplace. Show all step and calculations, and provide a written description of each step. Then do the same using different initial conditions of your choice.
2 Problem Statement Consider a simplified two-joint robotic arm where the two joints are connected by rigid links. Each joint has a mass, and the links are modeled as idealized rods. The motion of each joint affects the other through the coupling of forces. The system can be modeled by the following second-order coupled ODES: m₁Ö₁ + k₁(01 — Orest) + k3 (01-02) = 0 (1) m202 + k2(02 - Orest) + k3 (02 - 01) = 0 where 01(t) and 02(t) represent the angular positions of the first and second joints, respec- tively. m₁ and m2 are the effective masses at each joint. k₁, k2 and k3 are the stiffness constants representing the elastic restoring forces in the joints and links. Orest 0 represents the equilibrium position for each joint. 3 Project Tasks: = The parameters are: m₁ = 1, m2 = 1, Orest = 0, k₁ = 1, k2 = 2, k3 = 3, Orest = 0 The initial conditions are: 01(0) = 1,02(0) = 2 A written report that includes: 1. A description of the solution method. 2. Plots of the joint positions over time. 3. An analysis of the oscillatory behavior for another different initial condition that you like.
Transcribed Image Text:2 Problem Statement Consider a simplified two-joint robotic arm where the two joints are connected by rigid links. Each joint has a mass, and the links are modeled as idealized rods. The motion of each joint affects the other through the coupling of forces. The system can be modeled by the following second-order coupled ODES: m₁Ö₁ + k₁(01 — Orest) + k3 (01-02) = 0 (1) m202 + k2(02 - Orest) + k3 (02 - 01) = 0 where 01(t) and 02(t) represent the angular positions of the first and second joints, respec- tively. m₁ and m2 are the effective masses at each joint. k₁, k2 and k3 are the stiffness constants representing the elastic restoring forces in the joints and links. Orest 0 represents the equilibrium position for each joint. 3 Project Tasks: = The parameters are: m₁ = 1, m2 = 1, Orest = 0, k₁ = 1, k2 = 2, k3 = 3, Orest = 0 The initial conditions are: 01(0) = 1,02(0) = 2 A written report that includes: 1. A description of the solution method. 2. Plots of the joint positions over time. 3. An analysis of the oscillatory behavior for another different initial condition that you like.
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