I support Edward's suggestion. Although the SimmSpline can be differentiated to give velocities and accelerations, I would not recommend this. Interpolating splines are sensitive to noise and will give unreliable derivatives. To see an example, run my test code with these knots: x = 0:10; y = x + ra...

You're welcome! I am not sure it's a bug. The extrapolation always looks correct in my tests with SimmSpline. I have not thoroughly analyzed the code, but keep in mind that an array in C++ starts with index 0. With n knots, there are n-1 knot intervals, indexed from 0 to n-2. The polynomial with ind...

Splines is one of the topics in my numerical methods class this semester, so I was curious about the SimmSpline. It is a cubic spline, which means that it is a piecewise cubic polynomial; a different polynomial in each interval between two knots. A spline is smooth, which means that the function, an...

Mohammadreza, Thanks for the reference and links. I was not familiar with that paper, it's very interesting and provides a method to do static optimization with a compliant tendon. There still is no muscle dynamics, because the force-velocity properties are not used. It is a static muscle model, a s...

The faster computation that Nicole refers to, is related only to forward dynamics (simulation). Static optimization (as the term suggests) considers only static muscle properties and won't do a dynamic simulation. In the classic form of static optimization, it only needs the maximal isometric force ...

Ross, I can confirm that I often see this behavior in IPOPT and I have wondered about this too. It definitely seems wasteful to iterate like this. Maybe it helps IPOPT escape from a local minimum, but then you wonder if there is a strategy behind it. My earlier collocation work was with the SNOPT so...

I also tried to find the answers in the C++ code and got about as far as you did. Someone must know... The Boyd reference seems incorrect to me. This inverse kinematics problem is not a convex optimization. If computation speed is not important, it does not matter which algorithm is used. They will ...

David, These are good questions and it's good to see that you want to get to the bottom of this. In robotics, inverse kinematics is only concerned with the position/orientation of the end effector. If the robot has more than 6 DOF, this is an underdetermined problem, with an infinite number of solut...

Carlos, Thanks for posting those graphs. Your muscle activations don't look like the expected bang-bang control. You are probably aware of the old work from Stanford and Amsterdam on squat jump optimization [2,3]. These demonstrate a bang-bang control solution. The 50 rad/s bound on angular velociti...

You are right, the jump should be higher. I don't think it is the goals weighting. With less activation, the controls goal now has a lower value and that would still favor a higher jump. It would be good to plot the optimal controls. In a maximal effort task, the optimal control solution should be "...