Dear All
There is 1-DOF Series Elastic Actuator(the scheme and state equation is given in the image). A=[0 1; -K/m 0] B= [0 -1/m]*F
where K is the stifness constant of the spring, m is the mass of joint, x is the distance, done by spring.
Could you give me the suggestions on how to write the controller, which gives proper value of spring constant(K) according to the desired(reference) distance, which must be passed by spring. It is necessary to highlight that the control input force is constant and distance can be controlled only by modifying the value of spring constant.
My previous idea was to apply adaptive MPC controller(with Kalman Filter or System Identification).
But the problem is that the value of stiffness constant, is located in "A" matrix, and it must be treated as "INPUT" variable, and not as variable with disturbance or nonlinear one.
P.S. It also didn't work with Linear Function.
Thank You in advance.
Hi Akrom Magzumov,
Your model looks like a Maglev system. Anyhow, let's get into your mass-spring model.
m·x'' = − k·x + F.
For simplicity, let's make m = 1 and F = 0. Thus, we have
x'' = − k·x.
If you had studied eigenvalue problems previously, you would have known that if k is a positive constant, then x will result in a perpetual oscillations from the non-zero initial position.
Naturally, adding a damper would have solved the problem beautifully. Since it must be a spring-driven control system, then the stiffness of the spring (k) must be modified in a nonlinear fashion to embed the damping effect. ← Hint! 💡
Assuming that the mass-spring system can only operate in the range 0 < x ≤ xmax, and the input force is given by F = m·xd, where xd is the desired position at steady state, here are the results. Try to design a nonlinear function for k as an exercise.
Dear Dr. Yew-Chung Chak, beautiful solution. I appreciate that you have dedicated the time to help me.
One moment that I need to specify, the input force will be not given by me. And the value of the input
force is not determined by me, I know that it must be just constant(it is some value and it will be given). In this case, what is possible to do?(But I still can add the damper)
Yew-Chung Chak
Dear Akrom,
If you can add the damper in parallel to the spring in this case the pair (A, B) would be controllable through any input and you can apply an controller of your choice( PID, MPC…).
Best regards
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