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I've got a problem with my equation that I try to solve numerically using both MATLAB and Symbolic Toolbox. I'm after several source pages of MATLAB help, picked up a few tricks and tried most of them, still without satisfying result.

My goal is to solve set of three non-polynomial equations with q1, q2 and q3 angles. Those variables represent joint angles in my industrial manipulator and what I'm trying to achieve is to solve inverse kinematics of this model. My set of equations looks like this: https://i.stack.imgur.com/ZrWO6.jpg

I'm solving it with

numeric::solve([z1,z2,z3], [q1=x1..x2,q2=x3..x4,q3=x5..x6], MultiSolutions)

Changing the xn constant according to my needs. Yet I still get some odd results, the q1 var is off by approximately 0.1 rad, q2 and q3 being off by ~0.01 rad. I don't have much experience with numeric solve, so I just need information, should it supposed to look like that?

And, if not, what valid option do you suggest I should take next? Maybe transforming this equation to polynomial, maybe using a different toolbox?

Or, if trying to do this in Matlab, how can you limit your solutions when using solve()? I'm thinking of an equivalent to Symbolic Toolbox's assume() and assumeAlso.

I would be grateful for your help.

am304
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Krystian Meresiński
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1 Answers1

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The numerical solution of a system of nonlinear equations is generally taken as an iterative minimization process involving the minimization (i.e., finding the global minimum) of the norm of the difference of left and right hand sides of the equations. For example fsolve essentially uses Newton iterations. Those methods perform a "deterministic" optimization: they start from an initial guess and then move in the unknowns space essentially according to the opposite of the gradient until the solution is not found.

You then have two kinds of issues:

Local minima: the stopping rule of the iteration is related to the gradient of the functional. When the gradient becomes small, the iterations are stopped. But the gradient can become small in correspondence to local minima, besides the desired global one. When the initial guess is far from the actual solution, then you are stucked in a false solution.

Ill-conditioning: large variations of the unknowns can be reflected into large variations of the data. So, small numerical errors on data (for example, machine rounding) can lead to large variations of the unknowns.

Due to the above problems, the solution found by your numerical algorithm will be likely to differ (even relevantly) from the actual one.

I recommend that you make a consistency test by choosing a starting guess, for example when using fsolve, very close to the actual solution and verify that your final result is accurate. Then you will discover that, by making the initial guess more far away from the actual solution, your result will be likely to show some (even large) errors. Of course, the entity of the errors depend on the nature of the system of equations. In some lucky cases, those errors could keep also very small.

Vitality
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