Project $2024$

ssignments$/1443294$

The mathematical modeling of these systems allows engineers to predict the behavior of robotic arms and design control strategies to minimize undesired vibrations. This project will focus on a simplified two$-$joint robotic arm, which can be modeled using a system of coupled second$-$order ODEs.

$2$ 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:

$m_1{}_1^{}+k_1({}_1-{}_{\text{r}\text{e}\text{a}\text{t}})+k_3({}_1-{}_2)=0$

$m_2{}_2^{}+k_2({}_2-{}_{\text{r}\text{e}\text{a}\text{t}})+k_3({}_2-{}_1)=0$

where ${}_1(t)$ and ${}_2(t)$ represent the angular positions of the first and second joints, respectively. $m_1$ and $m_2$ are the effective masses at each joint. $k_1,k_2$ and $k_3$ are the stiffness constants representing the elastic restoring forces in the joints and links. ${}_{\text{r}\text{e}\text{s}\text{t}}=0$ represents the equilibrium position for each joint.

$3$ Project Tasks:

The parameters are:

$m_1=1,m_2=1,{}_{\text{r}\text{e}\text{s}\text{t}}=0,k_1=1,k_2=2,k_3=3,{}_{\text{r}\text{e}\text{s}\text{t}}=0$

The initial conditions are:

${}_1(0)=1,{}_2(0)=2$

A written report that includes:

A description of the solution method.

Plots of the joint positions over time.

An analysis of the oscillatory behavior for another different initial condition that you like.