# Free body diagram and equilibrium of a differential-drive mobile robot

I'm trying to study the free body diagram of this differential-drive mobile robot:

So i started considering just a straight motion (torque on the right wheel is equal to the torque on the left wheel) and i wrote down the equilibrium equations about forces and torques. For each active wheel i considered 5 forces acting on it (mg, vertical reaction due to the surface, friction force, horizontal reaction force at motor bearing, horizontal reaction force at motor bearing) and a torque generated by the electric motor.

$$x:m\ddot{x_A}=-(P+F_T)\\y:F_N=mg+P'\\J\ddot{\phi}=\tau+F_Tr$$

$$A$$ is the wheel center of mass, $$F_N = R$$, $$F_T = F$$. Since it must be a pure rolling motion:

$$F_T\le f_aF_N$$

I wrote the same equations also for the passive wheel, in which I did not have $$\tau$$ torque.

$$x:m\ddot{x_B}=-(P1+F_{T1})\\y:F_{N1}=m_1g+P1'\\J_1\ddot{\psi}=F_{T1}r_1$$

$$B$$ is the wheel center of mass.

About the platform I have something like this:

Assuming that the robot is moving on a straight trajectory with the same torque applied on each active wheel, I have this equilibrium equation about the x-axis:

$$m_c\ddot{x_c}=2P-P_1$$

From the equilibrium equations of the passive and active wheel, I got $$P$$ and $$P_1$$:

$$m_c\ddot{x_c}=2\left(\frac{\tau}r-\frac{J+mr^2}r\ddot{\phi}\right)-\frac{J_1+m_1r_1^2}{r_1}\ddot{\psi}$$

I have three unknowns and just and equation. Am I doing something wrong?

• Hello! I have edited your question using MathJax (LaTeX) math typesetting. For future questions, you can refer to MathJax basic tutorial and quick reference. Thanks! – Jonas May 29 at 21:28
• Oh, thank you. What about the dynamical analysis? Can you help me? – xWalle May 30 at 9:37
• @xWalle Better ask it on engineering stack exchange. You will definitely get someone to answer your Q. – Srijan M.T Jun 2 at 14:42