Nt1310 Unit 8 Study Guide

The coordinates of the system is defined by , θ = angle of the chassis from vertical, α = angle of tread assemblies from vertical,
Ø = rotation angle of tread sprockets from vertical, mc = mass of chassis, mT = mass of tread, ms = mass of sprocket,
Lc = length from centre of sprocket to centre of chassis,
LT = length from centre of sprocket to centre of tread assembly.

The kinetic energies of the sprocket, chassis and tread assemblies are given respectively , T_S=1/2[m_c x ̇^2+J_S φ ̇^2] (1) T_C=1/2 [〖m_c (x ̇-L_c θ ̇ cos⁡θ⁡)〗^2+m_c (〖L_c θ ̇ sin⁡〖θ)〗〗^2+J_c θ ̇^2 ] (2) T_T=1/2[m_T (〖x ̇-L_T α ̇ cos⁡〖α)〗〗^2+m_T (〖L_T α ̇ sin⁡〖α)〗〗^2+J_T α ̇^2] (3)

The gravitational potential energy is given by ,

V=m_c gL_c cos⁡〖θ+m_T gL_T cos⁡α 〗 (4)

The generalized coordinates are defined as , q〖=( x ∅ α θ )〗^T (5)

The Lagrangian is given by