Hi Walid,

A good starting point to address your issue may be Paccot, "Review on the Dynamic Control of Parallel Kinematic Machines: Theory and Experiments", 2009 available from http://www.irccyn.ec-nantes.fr/~martinet/publis/2009/IJRR09flavien.pdf

Complementary reading may be found by checking for:

• https://hal.archives-ouvertes.fr/hal-00401684/document
• https://hal.archives-ouvertes.fr/hal-00838470/document
• https://hal.archives-ouvertes.fr/tel-00777361/document

Best regards

Dear walid,

I suggest to you attached files in topics.

Best regards

you can go to ''  DYNAMIC MODELLING AND SIMULATION FOR

CONTROL OF A CYLINDRICAL ROBOTIC MANIPULATOR

By

ASIF IQBAL and S. M. ATHAR ''

If your question is about "Direct dynamics" then your goal is to find the joint accelerations starting from the knowledge of the initial conditions (in term of joint positions and velocities) and of the command torques/forces and also the external ones applied to the end-effector (of course time by time).

If you have already modelled your manipulator by means of the lagrangian equations, you should have a non-linear matrix equation involving all the joint quantities:

H(q) q'' + C(q' ,q) q' + g (q) = f

where q,q',q'' are respectively the joint positions, velocities and accelerations (gathered in a vector) , H(q) is the matrix that contains all the intertial terms, C(q' ,q) is the matrix that contains all the Coriolis and centrifugal terms, g(q) is the vector of the weight forces/torques and f is the vector of all the forces/torques acting on the joints (they may be both command inputs and transmitted forces/torques from the tool-center-point).

From this matrix equation you could derive a (non linear) state-space model simply by putting as state variables the joint positions and velocities: x1 = q , x2 = q ' so that your state equations will be x1 ' = x2 and x2' = -H-1 ( Cx2 +g) - H-1f.

Of course, your input shall be f as a function of time (which has to be known in this kind of problem) whereas your output can be for instance one of the two joint quantities.

DEAR MR WALID

HOPE BELOW ANSWER COULD HELP

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