TheSchatz Mechanism should move like this
However, with the default solver settings it froze after a few seconds in a planar link configuration. To make it run, I played around with advanced solver settings. Here is one attempt that went nuts:
(More solver settings for strange behavior can be found here:Schatz_Linkage.msim)
Some people might find this amusing. Of course, it is less fun when the initial plan was to spend an hour just for fun with a simple model (an hour is a fair estimate for similar simple looking models in MapleSim). The immediate reaction when seeing such simulation results is to blame the software for being either buggy or incapable. In this case, however, this was not the case, but identifying the root cause was not obvious.
The Schatz mechanism is a so called closed-loop mechanism where the links of the mechanism form a loop (the ground in the model closes the loop). In general, building and modeling mechanisms with loops is less straight forward than thought. Without a-priory knowledge or help (either by documentation or software hints) users can quickly find themselves in a situation of desperate trial and error. What was easy with other models can become a frustrating experience with unsatisfactory outcome. This happened to me on various occasions.
What makes closed-loop mechanisms more challenging? After resting for a long time on my virtual pile of unanswered questions, it turned out that the model, on top of being a closed-loop mechanism, is ill-conditioned: The Schatzmechanism is anover-constrained mechanism that is only mobile for certain geometric parameters. MapleSim can simulate such over-constrained mechanisms, but this can be a balancing act for the solver.
Who could have known this? A knowledgeable expert might say that users who do not know what they are doing should not use the software. But how to become aware of over-constrained assemblies when building and running a model in MapleSim does not require to be an expert? In this case the geometry was taken from a reference that sets the length of the ground link to Ö3. Model build, assembly and simulation instantly worked … but not for an extended time span.
In retrospect, everything is clear. Models that do not assemble do not fit together. Models that freeze in motion “jam numerically”. Linkages and joints of closed-loop mechanisms made of infinite stiff components may not fit together in all geometrical configurations. During runtime, after successful assembly, a stiff model can make a simulation sensitive to numerical errors. This does not mean that the user is dealing with a so-called numerical stiff problem that can be addressed by using stiff solvers. In this case, stiff solvers could not prevent sudden freeze or inversion of movements.
The only remedies that work for infinite stiff and over-constrained mechanisms are the ones that work also in real life. By either introducing mechanical play or elasticity in supports, joints and links, the simulation becomes robust. Numerically, for this case where none of the many advanced solver options made a difference, a simple increase of the relative error in the standard simulation settings worked. This remedy could be described as introducing morenumerical play. Interestingly in a completely different approach of animating a Schatzmechanism @one man also needed to introduce “deformations” in hissimulation to make it work.
The Schatzmechanism is of little commercial interest and can therefore be shared. Is it a rare case of successful assembly and freeze during runtime or is it more frequent that users run into similar problems? Only MapleSoft can tell, but in the latter case it could make sense that MapleSim supports the user. I see several possibilities for that:
- A more prominent mention in the documentation that kinematic loops require caution could raise awareness.
- Algorithmically detecting kinematic loops and informing the user that closed loops can be potentially over-constrained in certain geometric configurations.
- (If possible, analyzing the Jacobian in the frozen configuration might give better hints than solver messages during runtime can provide. The attached model gives the hint with MapleSim 2025.1 that the error tolerance might be too tight, but no indication why.)
- Implementing themobility formula, analyzing closed loops and issuing a warning when the mobility M is less than 1 (meaning no degree of freedom)
The latter option sounds appealing. However, the degree of freedom calculated by the mobility formula only provides a necessary but unfortunately not a sufficient condition for mobility. For example, connecting a prismatic joint coaxially to another increases the mobility by one but does not add to the mobility of a mechanism. This means that an advanced algorithm must take the orientation of joints into account to determine the effective degrees of freedom. On the other hand, the Schatzmechanism and some other mechanisms have a mobility ofM=0 but can be mobile for certain geometries.
Should Maplesoft implement mobility analysis or are CAD tools that offer some sort of mobility analysis more suitable? In my opinion, from a conceptual point of view, it would be beneficial and faster to have this support already in MapleSim before going into details.
Should the user refrain from modeling infinite stiff mechanisms? I do not think so because they are useful in the context of deriving (analytical) forward and inverse kinematics. Furthermore, there are more mechanisms out there that are mathematically, according to the mobility formula, immobile but useful in daily life. Thetelescopic fork is a prominet example.
Final note for math enthusiasts:
The Schatzmechanism (invented by Paul Schatz) is a byproduct of theinversion of a cube. Recalling that the diagonal of the unit cube isÖ3 gives a hint of why the Schatz mechanism becomes mobile for this parameter. Also related to the inversion of a cube is theoloid: a solid with a developable surface that touches with its entire surface a flat surface whenrolling. The oloid and the Schatz mechanism are closely related, which can be appreciated from thisvideo.
