Constraints other than nonnegativity on ode solver?

I'm using ode45 to solve the equations of motion of some robot arm. I know I can use the odeset option 'NonNegative' to tell the solver that certain states must be nonnegative, but what if I want certain states to be bounded by different numbers? So say, I have a state q: q = [q1 q2 q3] and I want q1 to be in between -pi/2 and pi/2 and I want q3 to be in between 0 and 1... Is there any way to do this?

Thanks in advance

Answer 1

[Matt J]

You could try transforming the state variable like

   q3=sin(y)/2+1/2

where y is the transformed state variable. The derivatives in the state equation also have to rewritten in terms of y

   dq3/dt= (cos(y)/2)*dy/dt

but it looks like the cos(y)/2 part can be accounted for using a mass matrix.

[Iris]

Hi, Thanx for your reply. I'll give it a try. Won't it screw up my equations of motion?

I don't understand though what you mean with the mass matrix?

[Matt J]

You'll have to try it and see what you get. However, I don't see how it would screw up the equations, because the new equations are derived by substituting q3(y) into the original equations. Therefore y can only satisfy the new equations if q3(y) satisfies the original ones.

The use of a mass matrix is an option is mentioned in the documentation for ode45.

[Iris]

Unfortunately, this doesn't seem to work. The point is, when z becomes zero, it should stay zero because the arm cannot drop through the floor. It should only go up again because there is a force making it go up. If I use a sine function, I do not get this result which kind of makes sense, as a sine wave is an oscillation. I really hope there is another solution to this problem.. Would be most appreciated!

[Matt J]

Doesn't the floor exert a force on the arm? And don't your equations of motion account for that force together with any others? If so, why do you need to add separate constraints in the first place? Why don't the applied forces alone (and hence the equations of motion) enforce the constraints?

In any case, did you test the final state trajectory q3(y(t)) that my approach gives you? Does it not satisfy both the equations of motion and the constraints? If so, and yet you think there is a 2nd solution out there that you like better, it means that the solution is non-unique. In turn, that means you haven't provided enough equations to fully specify the solution.

[Iris]

Normally I would add constraints to my equations, but this is an assignment for which I have to use the port-Hamiltonian formalism and I have not included the constraints. But I was just going to try to build in constraints now myself. I'll let you know if it worked. Thanx for your input so far though!