Dear Mahdi Nabipour

I think in the analytical dynamic approaches, the Hamilton approach which is base of the famous Lagrange method, can be used for obtaining forward dynamics equations of multibody systems. I have seen a paper months ago, maybe be useful for you.

Article A parallel Hamiltonian formulation for forward dynamics of c...

Dear Mahdi Nabipour,

Among the most popular methods you have Euler-Lagrange, Recursive Newton-Euler, Finite Element. For more information about this subject, i suggest you to see links and attached files on topic.

https://link.springer.com/article/10.1007%2Fs11044-005-1143-9

https://aaltodoc.aalto.fi/bitstream/handle/123456789/13940/master_Ruf_Oliver_2014.pdf?sequence=1&isAllowed=y

https://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-17592011000600002

https://www.liebertpub.com/doi/abs/10.1089/soro.2017.0079?journalCode=soro

https://hal.inria.fr/hal-01649355/document

Article A review on kinematic analysis and dynamic stable control of...

Article Modeling and Simulation of Robotic Systems with Closed Kinem...

Best regards

Dear Mr. Nabipour

Hello

I don't think this system is underactuated.

Second, the motion of the cart is constrained to the circle and this imposes only one constraint equation. If, in fact, the motion of the cart was constrained with two constraints then the cart won't move (grounded).

The constraint in the system implies that the cart should always move on the circular track. In velocity level, this means that the velocity of the cart in the normal direction is zero. Based on the constraint, the force that keeps the motion is normal to the track.

The other force component, i.e the tangential component, is not a constraint force but a friction force that depends on the normal constraint force.

I would suggest the constrained Lagrange method. I have almost worked with all the analytical approaches, constrained Lagrange is a simple yet powerful approach.

I have included a paper which will help you a lot.

To sum up, each link has one kinematic parameter and the system has only three actuators. But, since there exists one constraint in the system, the whole system is a fully actuated 3-DOF system.

I have analytically modeled the equations on paper (attached).

Also, I have derived the equations with a symbolic toolbox and finally, after forming the state space, have solved the equations (attached).

Care should be given to the initial conditions since the kinematic positions and velocities are not independent.

I have also checked the constraint and energy errors to make sure everything is alright.

Suitable codes for the animation of motion are included.

Finally, you can develop and add any control policy, include it in the simulation, and see the results.

Best luck,

Lagrangian mechanical systems subject to unilateral constraints, impacts and friction, make a rich class of nonsmooth, nonlinear dynamical systems. Though a lot remains to be discovered, it seems that constrained manipulators, biped robots, kinematic chains with joint clearance, juggling systems, tensegrity (cable-driven) and tethered systems, etc, possess different stability and control properties. It is therefore necessary to analyse the control of each subclass separately. Within the framework of multi-rigid-body systems with joint clearance, the main modelling issue concerns the choice of the contact model. Clearances involve additional degrees of freedom and introduce unilateral constraints. There are several available classes of models for collisions between bodies in the Mechanical Engineering literature