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.
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 to get θ1(t) and θ2(t) using eigenvectors and eigenvalues. Show all steps/calculations, and provide a written description of each step. Then do the same using different initial conditions of your choice.
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