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: m1 ̈θ1 + k1(θ1 − θrest) + k3(θ1 − θ2) = 0 (1) m2 ̈θ2 + k2(θ2 − θrest) + k3(θ2 − θ1) = 0 (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 provide a written report that includes: 1. A description of the solution method (show for all steps/calculations) 2. Plots of the joint positions over time. 3. An analysis of the oscillatory behavior for another different initial condition that you like