Usage and Functioning of the LDNFS Simulator

Part 3/3

[ALMOST READY DEMO VERSION: SO, TO SAVE PAPER, IF POSSIBLE, DO NOT PRINT IT UNTIL IT'S FINALIZED ]

what am I reading? What is this text, webpage or book? - abstract:
This is the Part 3/3 of introduction and guide to the use and functioning of the Lagrangan Dynamics' New Flavour Simulations computer-software. Designed to simulate most types of 3D articulated bodies ; based on an enhanced flavour of the Lagrangian Dynamics as a theoretical framework specialized in simulating and studying the motion of various, actually most types of, articulated mechanisms.

A guide, which serves not only the purpose of enabling to basic as well as advanced use of the Simulator, but as well that of providing a detailed introduction to the new flavour of the Lagrangian Dynamics which underlays the Simulator's functioning. Where, since the new flavour of Lagrangian Dynamics enables to do practically all what the classical flavour enables to, and even more, this guide qualifies also as a complete introduction to Lagrangian Dynamics itself. A very friendly and easy introduction since the new flavour of Lagrangian Dyanmics is itself a bit simpler, friendlier and easier to understand than the classical flavour of it: a feature of it which I have researched for, intentionally... to allow making the powerful and fantastic method of generalized coordinates more popular and easy to use, instead of just constructing a simulation-software with it. So, Enjoy!
Brought to You by Simon Hasur (a.k.a. The Nerd of Algorithms)

Summary of part 2/3 (optional)

Essentially, I've picked up for You 4 facts which support the idea that the Lagrangian Equation of Motion is nothing so bad or complicated ; even if it is a so-called partial differential equation (usually stated to be a bad thing but which it is not int this case), either a single one, or a system of them, according to the situation.
These 4 facts give You a very solid base when dealing with this equation ( or a system of them). So that, if any doubts arise, You can grasp these 4 facts and hold them as stong as needed, to dissolve any substancial doubt.

fact n.1 )

We wrote down the Lagrangian Equation of Motion as
;
the most common way it's written on books and tutorials on Lagrangian Dynamics.
Where

where a generic xk and Fk is to be interpreted as illustrated by the graphical reference givel below.


Very fine. When we have more than 1 degree of freedom, we have a system of as many of this equation, as many degrees of freedom our mehanism has.
Where, how many degrees of freedom ( or DOFs for short) a mechanism has, is simply the number of parameters - called generalized coordinates - needed to define univocously it's configuration at a given time.

Once familiar with this compact way of writing the equation, we set out to analyze a bit it's structure. It looks a pretty bad differential equation, but our task now is to realize that it's not so bad.
As first step, we rewrite it with the d/dt operation developed fully, to get:
;
often written also more compactly as
;
where is either case, all that sort of total derivative, that summatory with the j index, comes out because of the fact that both q and qd, actually are function of t as
q(t) and qd(t)
which forces to do a depth-2 composite function's derivative with respect to the t variable. To give the above-reported formula.

The out-developed form, for us is certainly the most practical for us, and leads directly to a nice matricial notation like this,
,
as soon as we collect the qdd terms and express the sum as a typical matrix-vector product as
A*qdd
and then we pick the term with the qd and express also that as a matrix-vector product as
B*qd
and leave at right what was at right, and call it
k (as in the above equation), or b (as it is called throughout the rest of this book), or whatever You like ; a vector made of all known terms.

So, our equation is nicely rewritten as:
A*qdd + B*qd = b ;
here, we could already stop since as soon as we bring B*qd at right and leave b at right and sum it into one vector, we have the qdd explicitely, since it's a simple linear algbraic system, from which we can pick it out since the qdd is the vector of solutions. Simply by running the Gaussian Elimination procedure, for example.
So, fact number one is that the Lagrangian Equation of Motion is not so ugly because it's very very friendly for implenting with it a numerical simulation.
Technically said, the Lagrangian equation of Motion is linear in the qdd term: more on this later if You are interested in it, in the section called How does the Simulator work ? .
Meanwhile, before closing this paragraph, let's clarify a bit why in the context of a numerical simulation, this fact of treating the Lagrangian Equation of Motion rewritten as
A*qdd + B*qd = b
is legal.

This is discussed in detail in part 2/3, but we can summarize the main facts, in short, here. Since the matrix-vector products of the form from which started, suggest that all the partial derivatives in the Lagrangian Eqation o Motion concur to make up sorts of total derivatives, clearly they make up some avarages at each simulation-step: so that for a numerical simulation, it's sufficient to just evaluate each of those partial derivatives and consider them constant for that small timestep ; or do a multistep-evaluation like the Runge-Kutta numerical method suggests. This is all legal, said in one sentence, because the relation between the partial derivatives among themselves and their relation with the qdd(t), qd(t) and q(t) terms, is much weaker than the relation between these ; and don't forget that the partial derivatives, as they concur to sorts of total derivatives, are in fact avaraged. Moreover, in simulating a linear dynamical system (discussed later), all the matrices holding the partial derivatives, reduce to constants instead of being results of the evaluation of functions of q and qd ; so this is even more evident. To conclude, we managed to give also a physical justification ; and not only a mathematical one. Hope that You appreciate!

fact n.0 )

Very fine. Let's pick up some other points in favour of the friendliness of the Lagragian Equation of Motion.
Well, it's all constucted around the T() or L() function, which is simple becase:
a) it's a scalar function.
b) the structure of this function is very characteristic and simple, that is,
,
as proven by my first theorem on Analaytical Mechanics. Let this fact be our point number zero in our list of facts.
In fact we also see that instead of a generic
fabc( q ) function
we can quitely restrict that to be a nice polynomial,
Polabc( q ) function.
To arrive at the heart of our Simulator's reliable functioning, the polynomial formulation of the Lagrangian Equation of Motion:
[INSERT FORMULA]
which, as we will outline it quickly in the introduction to the part 3/3 of this guide, has all the nice properties that a numerical simulator such as ours, needs to function 100% well in all allowed circumstances.
Now, let's go back to the structure of the Kinetic Energy Function: to illustrate a bit more clearly what it's structure means. In one sentence, it means that the structure of it is the seme, and it's formally a sort of generic square like
( ax + bx + cx + dx + ...)2 + ( ay + by + cy + dy + ...)2 ( az + bz + cz + dz + ... )2
but with all the coefficients of the final result being a custom value. A custom value depending on the q vector of gneralized coordinates, according to
Tk = 0.5*mk*(xd( q, qd )2 + yd( q, qd )2 zd( q, qd )2) .
It's vaguely similar in structure to the kinetic energy of the single free material point in the context of Newtonian dynamics,
T = 0.5*m*(xd2 + yd2 zd2) ,
but it's more generic. Because there are not only the squares of each of the variables, but also the products of all the combinations of variables which there are in the expression. As each of the squares in the prevous expression, are squares of polynomes made of an expression of the q values, moltiplied by one (!!) of the qd values. A consequence of the definition of the derivative, with repect to time, of each of the expressions for each of the 3 coordinates, of the position
xk ( q )
of each material point constituting the dynamical system to simulate. It's a fact deriving from the simple application of the Chain Rule for multivariate functions, of the form
f( q0(t), q1(t), q2(t), ..., qN_DOF-1(t) )
when performing it's derivative wih respect to t:
d f( q0(t), q1(t), q2(t), ..., qN_DOF-1(t) ) / dt ;
or written more compactly,
d f( q(t) ) / dt .

I've prepared for You a more concrete and procedural defintion for the Kinetic Energy Function ; which states formally the seme thing as what I've tried to outline on the last few lines.
Before showing it, let's introduce it... .

So. By looking at
( a + b + c + d + ...)2 ,
it's clear that whatever coefficients we put before each term of the result, ( a2 + b2 + c2 + d2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd + ...),
it always has the seme structure. So that each of the 3 components x, y, z
T = Tx + Ty + Tz ;
of the the kinetic energy function of each material point of a dynamical system ; is always the sum of a pair of 2 functions:
1. an (N_DOF+1)-dimensional paraboloid (or ellipsoid? - it may have different names, but it's him) (2 DOF example in the fugure below ) ;
,

2. a (N_DOF+1)-dimensional hyperboloid (2 DOF example in the fugure below ): for this one, I've prepares 2 graphics because it's shape is a bit tricky.
,



Where, being clearly the kintic enegy function always positive for physcal reasons, that is because each matrial point of the system is the result of square and so can not have a Kinetic Energy vlause minor than 0, the sum of the 2 function-prototypes listed, is always a paraboloid directed upwards (figure below ).
;
where in this last graphic, note that it's still a paraboloid but it's axes of symmetri are not parallel anymore to the axes of the cartesian space.
A basic knowledge of modern multidimensional geometry allows You to imagine this in the generic case of the Kinetic Energy Function of a dynamical system with N_DOF degrees of freedom.
Enjoy!

The formal definition is reported below.
[FIGURE]
It's form should not come as a surprize in the context of what has been oultined before. It's intendedly but also factually resemblant of the simple 0.5*m*(xd2 + yd2 + zd2)
expression, but it's quite a bit more generic as one could expect.


And this is valid both in the classical as well as in the New Flavour of Lagrangian Dynamics. As an example, look at the Kineric energy function of the classical double-pendulum...

as it is in the classical flavour of Lagrangian Dynamics... this structure is evident in the qd1 and qd2 terms (thich here are called theta_d1 and theta_d2). The structure is this:
a2 + b2 + 2ab ;
which is the result of
( a + b )2 ;
whose graph is below, from 2 differen shots.
.
Where it is crucial to note that although the graph is always greater then or equal to 0.0, it's the limit case! For any a*b term with a coefficient greater than 2.0, the graph is like a hyperboloid ; while if it's minor than 2.0, it's a praboloid.


The kinetic energy funciton is a scalar function, R^2*N_DOF - > R^1 ... that means... a scalar ; that shuld be completely clear by now. But before goind on, it's in place to clarify that clearly in case of linear dynamical systems, the stuff ends here, as coefficients are constant. So for those interested in this, the question of how the paraboloid graph of the kinetic energy is in the N-dimensional case, it just a matter of applying some basic notion of multidimensional geometry. While for the N-dimensional cas for non-linear dynamical systems, well, the structure of the function as it depends ofn the qd vector of the rates-ofchange (in time) of the elements of the q vecotr of generalized coordinates, is the seme. But to fully visualize or just study a bit the structure of the kinetic energy funciton, we must take into account also th dependence of it on the q vecot and not only the qd vector of the rates of chenge of its elements. Well, one can choose mixed pair of axis, like a q1 and qd1 pair and so on all the various combinations, and it's fine... while to to it in general, one has to fiddle around a bit more heaviy with some basic concepts of multidimensional geometry : this is because in geometry, surfaces don't generalize so simply as parametrized curves.... while parametrized curves are always parametrized curves in 2D as well as in N dimensions, and one could plot in into one of those fancy hypercubes wich one can visualize form various angles with many programs on the internet ; with surfaces one has to cheat to get something meanigful. Think to the 2D graph of an R -> R function (not a 2D prametrized curve!) : it's simple... but it's generalization to 3D is a surface... and look at how more tricky their graps are! - this is because the 3D to 2D projection in this case is a bit chaotic. So, to graph and imagine a 4D surface or then an N-dimensional surface, it tends to get even worse. Sections can be considered, but this is not a book on multidimensional geometry, nor on scientific visualization, so let it end here. The material given on the topic so far is by large sufficient to get along with anything related to the structure of the kinetic energy function.

So, essentially one just has to keep in mind that in the case of a kinetic energy function such as in the example above, it's certainly true that the coefficients are custom, and true also that in non-linear systems, they are all functions of the q vector of generalized coordinates. But also that the coefficient for the mixed-terms (the non-square terms such as the a*b term of out previous example) are ALWAYS MINOR than 2.0 !!
Think also that: a dynamical system must always know where it's configuration should evolve in the next instant... an evolution whose direction is given by a sort of minimum of that parabolid stuff, that is the Kinetic Energy Funciton in our case, outlined above.
And this is clearly true even if we don't explicitely prohibit the use of composite functions in the parametrization, which, however is advisable.
In the polynomial formulation, clearly the use of composite funcitons is prohibited as it's simlpy impossible, and the corresponding expression for the Kinetic energy function (around varius configs), is left as an excercise... and it's solution is directly provided amond e sample-simulators made with the LDNFS Simulator.


Maybe this point zero of our list of facts seemed a bit theoretical to You, but... look, I explain it: I had used the Simulator throughout years without being yet able to justify formally why it works well and never fails. But now that I became able to justify it, I did it for You so that You can be really comfortable.
In fact I created it's first prototype back then, to avoid the very ugly failures of the previus naive version which did not restrict the work to the polynomial thing: I found no material whatsoever on why excactly it happened, and so deviced the heart of the New Simulator, that is my first theorem on Numerical Methods and the one on numerical stability which, however is only to keep the numerical simulation as close as possible to an ideal case of very small timesteps and lots of significant digits in all those variables used in the simlation which should behave as ideal real or rational numbers ; But theorem 1 I had worked out by pure intuition and rough exclusion of possibilities... I knew practically nothing of mathemetics then.
But now, at least I can bring You a more thorough justification of why it works always well, and what this means mathemeatically: So that You can be optimistic and not as desperate as I was when the old simulator used to come out with very ugly failures apparently in unpredictable circumstances. The New Simulator doesn't fail: and that's why I bring it to You, togather with this guide to it's usage and functioning.
You will read some more on these things in the introduction of part 3/3... meanwhile just take it as a guarantee that You are on a safe ground.

fact n.2 )

The next point, point 2 of our list of facts, is pretty simple.
Well, it's the very form of the solution of the Lagrangian Equation of Motion. It's a plain, very simple parametrized form, and there are practically no such things as initial conditions and such, since they are that, what intuitively there is, concerning a mechanism whose motion is one's intention to simulate. All what is needed is: an initial configuration, and an initial rate-of-change (in time) of each of the generalized coordinates. Pretty much it's all here. So, it's certainly very simple.
But if we want to dig into it a bit, let's state the thing a bit more rigorously.
We consider the 2 typical scenarios: the analytical solution, hypithetically ideal a, and a typical numerical solution.

Well, in the analytical scenario, once given the vector of initial conditions,
[ q0 ; qd0 ] ,
that automatically selects - so to speak- he right solution-set of out differential equation. And that selection pick up a completely univocously-defined solution.
Of course we don't have to forget that the analytically tractable cases are a minor subset of all the universe of practical mechanisms - most of which can be simulated only numerically... even if the concept remains the seme. We do a phase-plot (it has an ungly name, but don't worry), just look at the graph...



and can see that the initial cnditions select univocusly a solution-set. Some may display sorts of colosed-circuits, some may not... whatever ; it does not matter. For an 1 DOF sysem, we can plot the q on one axis of a 2D cartesian graph, and the qd on another. for a 2DOF system, we can plot q0 on one and q1 on one, and then do another plot where we put qd0 qd1 on the 2 different axes. Now, for the needs of applied things, the preference in case of a 2 DOFs system, is for the q0 and q1 case... because studying the evolution in time of configuration of the system is them most concrete tast. It' a parametrized curve, where the parameter is t (time, in our case) ; in every case.
In the case of generic N_DOF systems, in fact we have a parametrized curve living in an N_DOF-dimensional cartesian space: that is, simply an N_DOF-dimensional parametrized curve... as simple as that.


Instead, in the case of numerical simulations, which is here at the center of our attention, the situation is that, after the simulation has proved to be acceptably reliable after confrontatin with real-world measures or just the simpler pretention of a beliavable motion of an articulated mechanism, we just have to plot the q and qd variables on a 2d cartesian space... but that does not say much in practice as outlined before. So, let's take a 2-DOFs system, and see what a typical situation similar to the one outlined in the previous situation, would look like:
,
not much different, but things here are not so ideal. But as soon as we get a bunch of curves made of reasonably short segments after running a simulation with a bunch of different initial conditions, we can say that it's all fine. A fact which is, in most cases, also the sign that we did, if not a super-realistic, but at last an acceptable or at least a believable, simulation. That's normal, but now the question is that once we see the similarities in systems tractable both aalytically and both numerically, we can get rid of the consideration of the analytical case, as most practical articulated sysstems are not tractable analytically at all.
So... let's just generalize the numerical situation, and close the discussion.
So. For systems with more DOFs, cleary we always end up with a multidimentionsla parametrized polyline. There's nothing more to say... what in a simple, analytically studyable system proves to be a closed curve, in a numerical simulation is just almost close... it's similar and should be so, but that's it. The reliability of the simulation, in the case of generic polunomical systems, depends on 2 parameters: the value of the stepping-time dt, and the value of the small h value used to numerically get the finite-difference derivatives of all kinds needed, in the simulation-procedure.
As simple as that.

fact n.3 )

Now instead let's go to the last point of our list of facts, fact number 3: the form of the equation itself. This is the deepest part of it, so if You are tired, take a break here before going on.


So: point 3... let's go.
First, we analyze a bit the dependencies which there are in the Lagrangian Equation of Motion. According to my first theorem on Analytical mechanics, because of the structure of the T() or L() function, we have these dependencies:
A( q )*qdd + B( q, qd )*qd = b( q, qd ) ; where the most stricking fact is that A depends only on q, there is NO qd term in it, and moreover, it's symmetric.
From here, we write, to be sure also the dependences on the time's variable t
A( q(t) )*qdd(t) + B( q(t), qd(t) )*qd(t) = b( q(t), qd(t) ) ; and step to the form which immediately leads to a stepping numerical method:

qdd = A-1N_DOFxN_DOF( q, qd ) * b( q, qd ) ;

finally, not even so bad. It is immediate to implement a numerical simulator with it. Yes, that we knew already. But let's come to the last point, to total more-less 4 points to support that the Lagrangian Equation of Motion is, after all, not so ugly.
We analized the dependencies. But we remained a bit in surface. So, let's go deeper:
If we have a linear dynamical system, the B term is all-0... it vanishes. And the A reduces to a matrix of contansts, while instead b reduces to a vector of contants if we use T(), and a vector of inear expressions of the q vector. Familiar?
Yes! - it's a linear system of differential equations ; whose prototypic form is this:
;
which can always be re-written in this form relying on matrix-vector products, like this:

as it is commonly written on math books.
But since we are not being mathematicians here, we directly adopt the standart notation with q0, q1, q2,..., qN_DOF-1, and rewrite it as:
;
Which, with a practical compressed notation in the case of a generic N_DOF, becomes
.
Where, to be complete, if we want to explicit fully the dependence on t of all the variables in the q vector, we would write it as follows:
;
where for the generalization to the previous notation for generic N_DOF, requires only to replace - in that equation - qdd with qdd(t), qd with qd(t) and q with q(t) .


Hope this clarification was useful rather than confusing. Don't worry if anything is unclear... it's not essential to understand this for the usage of the Simulator.

Instead for those who aim at becoming advanced users of it or aim at becoming experts of Analytical Mechanics' new flavour, or also of the the classical flavour of it, well, we are here to support that purpose. It still may seem an annoyance, but in that case it's very important to become familiar with this notation, because it shows any linear sys of diff-equs in a prototypic, fixed-scheme, general form.

So, now that You know for sure what were we driving at when we wrote
A( q(t) )*qdd(t) + B( q(t), qd(t) )*qd(t) = b( q(t), qd(t) ) ;
we can now say that A is constant, B vanishes, and b is a vector containing one constant plus optionally some linear expression of the qi values... let's rewrite b as a matrix-vecotr product, of the form D*q. That's it.

Now we have
A*qdd(t) = D*q(t) + Fq ; a second-order linear system of differenial equations, or call it anyway... it's him, impossible to exhabge it with anything else. We can add a term which with funciton-constants of t but for that it's sufficient to let Fqi depend also on time. Fine, let's do it!
A*qdd(t) = D*q(t) + Fq(t) ;
But we are still there. Something is missing... .
It's linear, yes... but there is no term multiplying a qd, and Fq is a vector of contants which falls in the category of function-constant so that's fine.In fact the B term is non-zero only in non-linear systems, which in our case are all polynomial systems. So? If we produce expressions summarizing the effect of viscous frictional forces as functions of the qd vector (because of linearity it can only be made of contants) and so add a term for this in the equaiton:
A*qdd(t) + C*qd(t) - D*q(t) = Fq(t) ; to write it in conventional form with the function-contants at right.
This to show You that linear systems are so special also because, in order to -so to speak - feed it into a numerical solution program, all that would be neeeded is to give the A, C, and D matrices, and let Fqi remain interactively given or a vector of contants or whatever. So that these matrixes provide a true data-structure to contain, to define, a linear system.
This makes a lot of contrast with geenric polynomial systems, since there we have a much more subtle situation, which now we let in the background. As discussing it, would bring little-no practical advance.

Let it be sufficient to say that in the polynomial case, if we write out everytging explicitely, we ge to the form
A( q )*qdd(t) + B( q, qd )*qd(t) + C( q )*qd(t) - D( q, qd )*q(t) = Fq(t) ; where for major clarity I did not write explicitly the dependence of q ad qd on time when indicating on which of them each matrix depends... this to stress the fact that those matrices' elements are simply formal functions of q or qd, or either both. Anyway, for the seek of completeness I report also the full form where every dependence is explicitly written:
A( q(t) )*qdd(t) + B( q(t), qd(t) )*qd(t) + C( q(t) )*qd(t) - D( q(t), qd(t) )*q(t) = Fq(t) .
Where in either case this all derives from the seme formal re-writes of the Lagrangian Equation of Motion according to (figure with formula) ;
[FIGURE WITH FORMULA: lagrangian with d/ dt developed explicitely.]
where all the elements of all the matrices A, B, C and D in the previus form, are obviously, in general multivariate, polynomials. At least, we are on a very solid ground. Those interested in it, can easily figure out what kind of datastructure that polynomial form would need for it to be contained... interesting theoretically, but of little or no use in practice. All that reguards this and those things we call in this book polynomical dynamical systems, is summarized by my first theorem an Applied Mathematics (branch:differential equations): something not discussed here, but on another book which I shall write to enable advanced users to upgrade the simulator to simulate special dynmaical systems mentioned at (*1) in the chapter on parametrization rules, detailed discussion.


At least we have been as complete as we could, in the little space dedicated to the topic... since I prefer dedicating more to the application now than to the theory. After all, I am here to explain to You how apply the computer program based on the theoretical foundations I took care of: so that others don't have to do so, instead can apply the results immediately for some practical good.
Substancially, we have to abandon the compofortable ideal case of linear systems, and can not take advantage so efficiently of the concept of datastructure holding our system of differential equations: but we can still arrive at some very important and useful considerations. Let's go!
So. If we give up linearity and pass to polynomial case, the B term compeares with utter dependences both on q and qd (in an all-or-nothing style), and both A and C start to show dependencies on q, while D shows it both on q and qd (and has a separate contribution from T(), a diagonal matrix rather ill-manipulated, in fact an abuse of notation, and a contribution from the V() potential energy function if we use the equation of motion with the L = T-V form)... to get to our generic form discussed previously:
A( q )*qdd(t) + B( q, qd )*qd(t) + C( q )*qd(t) - D( q, qd )*q(t) = Fq(t) ; now written in conventional form with the function-contants at right.
So, we can assert that all dynamical systems, in the context of the Lagrangian Method, classical or new flavour whichever, all dynamical systems are literally like constructed, built, around a linear core - so to speak.
We know, linear systems never fail in simulation... they have, per definition, no of those crytical configurations (discussed below), as well as neither polynomial systems have them.
The core is always there, but with more intricate dependences instead of simple constants in the matrices. So, the A term is like a sort of one-man army surrecting the poorest cases: linear dynamical systems.
But the core is there, and A is always an invertible matrix, fact absolutely enforced for polynomial systems too.
But once understood this fact, for polynomial as well as all generic non-linear systems, the C and D matrixes become total abuses of notation and thus, once this analysis is over, return to the form:
A( q(t) )*qdd(t) + B( q(t), qd(t) )*qd(t) = b( q(t), qd(t) ) + Fq(t) ;
From which, since we are anyway in the context of a stepping numerical solution of this, we can leave comfortably out the dependence on t of q, qd as well as qdd and undoubtly of Fqi which in a simulator usually summarizes frictional forces and interactively-applied forces from simulation-step to simulation-step, to obtain: A( q )*qdd + B( q, qd )*qd = b( q, qd ) + Fq ;
fine.

As a finel note, as You could expect, this would not be so smooth with generic non-linear systems, where the invertibility of A could fail if the system entered a so-called crytical configuration... which is a problem with the prametrization, not discussed here since it was totally eliminated from the new flavour of Lagrangian Dyanmics. In fact in polynomial systems, there are no such crytical cofigurations. In fact the possible failures of generic non-linear systems are cast completely away from the New Flavour of Lagrangan Dynamics since it restricts to deaing with polynmial dyn systems exclusively... even though without breaking the generality which it achieves by requiring the chaining between the pairs of siulation-steps, or, conceptually, between polynomial parametrizations of the seme system, parametrization whose form remains always the seme, but requires to adjust some constants to when a mehcanism move outside the limit of displacements within the limits of mid-oscillatory motions.

Usage and Functioning of the Simulator

Introduction

Now. I have been in truble finding out a way to introduce You enough gradually, but without letting out anything important, to the parametrization techiques and strict rules which have to be respected when feeding into the sourcecode of the simulator, the definition of the articulated mechanism which is one's intention to simulate numerically.
Strict rules have to be respected because the simulator assumes that we are using those so-called polynomial systems. It assumes, as a hypothesis, as a definition, a basic fact, that our articulated mechanism to simulate is parametrized so as to lead to a so-called polynomial dynamical system - which is, as stated in part 2/3 of this presentation, a fantasy-name I gave for these types of systems because while dealing with them, progressively I realized their importance at least in the context of numerical simulations of articulated bodies and to minor extent also electromechanical systems ; and their importance notwithstanding, I never found any explicit reference to them as far as I had been searching, in books and the Internet, on Analytical Mechanics, Numerical Methods, or similar things. So that we can now, at least, refer with a unique name to a type of system which has all the properties needed for a reliable numerical simulation. So that we have now a proper name for - let's say - our favourite types of dyamical systems : polynomial dynamical systems.

This is the assumption: that the parametrization leads to a polynomial system and that thus the simulator works as expected, and works well.


Forgive me if I am a bit reluctant to start quickly, but very special care must be observed here. Becase on any reference-material I've seen so far on Analytical Mechanics, only 2 kinds of systems are taken into account: linear and non-linear dynamical systems... and there is no farther distinction. And as we'll see when we start introducing the parametrization-techniqes for doing the 4 most basic joint-mechanisms which I intend to discuss here, and which cover practically all the needs for simulating a practical articulated mechanism - whose destination, to say, is an implementation with a metal-construction, plastic-construction or similar - only 1 of them, the PRISMATIC joint leads to a linear dynamical system, while the others, according to the traditional classification, would lead to a generic non-linear dynamical system. But in the New Flavour of Lagrangian Dynamics underlaying the Simulator, these 4 lead to polynomial dynamical systems.


So. That's the reason why I'ts primary now for me to outline that the first type of articulation (and also a subtype of it), the PRISMATIC will be the first to be introduced ; because if only prismatic joints are present in a mechanism, that mechanism leads by itself to a linear dynamical system (the classical linear dyamical system we are well familiar with). So the parametrization of this type of joint will need little or no farther clarification. Even because it is well discussed on any good and detailed reference on Lagrangian Dynamics (or Analytical Mecahanics as it is also often called). So far, so good.

But I decided, and I hope it will prove the lucky decision, that before starting the discussion on how to parametrize the kind of joint which is the PRISMATIC joint and some other similar types of joints which behave similarly, before even starting, in order to make clearer the transition to other types of joints which don't lead to linear dynamical systems, I decided to go on here with some notes about, with some notes which will meke this transition less abrupt and much clearer.

So. We know that linear dynamical systems are usually very simple, and this is good (for those who are interested in this, also finding the analytical solution to the corresponding (ordinary) differential equation or system of them is a matter of play... there's a technique called Laplace Transform - applied to ordinary linear differential equatios or systems of them - which lets solve them algebraically... a techique and underlaying theor which - finally (!!) - gained some fame thanks to it's applications in various modern industrial fields ). But we also know that those mechanisms which qualify as linear dynamical systems, can not cover, even by far, all the prectical needs of a modern industrial world. And not even the needs of a fully complete theoretical framework to do articulated-body simulation. And so, we need also other types of joint-mechanisms, other than the prismatic or similar ones. And we must be able to simulate as well systems in which a rigid-body can also be totally free: expecially as reguards its orientation, since for it's position, more precisely the position of its conventional geometric center, it's as if it were a simple material point.
In our examples we saw similar cases.

So, a completely free body: free to go around, free to translate ; and also free to have any orientation... - the classical free body.
Or a similar case in which one of its points, which one can choose arbitrarily, is not free to translate around, but is tied to a fix point in a global reference-frame, while it's orientation can be any [min 8:13] ; it can change freely. Then one can attach also spring to it, and so on.

Of course we want to reach total generality.

We want that theese bodies, which are in most cases components of more complicated articulated mechanisms made of many rigid-bodies, can have any orientation.
And clearly in the classification of the linearity of a system - as I've already tried to advice about this - in the context of this simulator, we consider the Kinetic Energy function
T( q, qd ) ;
So that the dynamics of the system has to be linear for the system as a whole to qualify as linear ; and the Potential Energy function
V( q )
is not so important for us, since it reduces to a contribution to the rest of the external forces acting on the system, eventually leading to a set of generalized forces, as many as many degrees of freedom the system has. So that the Potential Energy function is just a contribution accoding to
Fqi = -dV( q ) / dqi + contribution of frictional and interactively-applied forces
If this troubles You, consider it to be polynomial too, or even linear: as for springs it's a always a good approximation even when one end of the spring is not moved along the direction of the line crossing it's 2 ends.
However, there's a reason to classify a system according to the Kinetic Energy function, a to let in the background the Potential Energy function. Mostly because we start designing a system by defining it's geometry ; how it should move intuitively, and not by putting some spring into the void of an empty reference-frame.


So, after discussing the simplest kind of joint-mechanism which leads to a linear system so that the parametrization rule to do it is the seme as in the classical approach (so that You can find lots of good references on books or on the Internet). Then we'll transition to te other 3 basic types of joints and the case of the free rigid-body, so that we'll have the FREE BODY, UNIVERSAL JOINT, SPHERICAL JOINT and HINGE JOINT, don't be surpreized when initially we will consider only situations in which the bodies are limited only to a small change in orientation around an initial orientation consiered as default.

And and after having discussed these cases, in which we will consider, in other words systems which exhibit only middle-oscillations (similar to small-oscillations but allowing considerably bigger motions and having a clearly much more complicated, colorful, dynamics than simple linear systems), so being all in general polynomial systems (discussed somewat in detail part 2/3 of this guide, which You could have skipped if You were not interested in it at that moment) ; and then we will, at the end learn how to chain these small- to middle changes in orientation of the bodies which can change it, to allow any change in orientation for the bodies in which need it in the system they contribute to. So, at the end we will reach full generality and be able to define and simulate any practical mechanism with full functionality of all the free bodies in it or bodies which are tied to some other body or a fix point with a UNIVERSAL joint, a SPHERICAL joint or a HINGE joint.


So, we initially restrict our work to mid-oscillatory systems: because these lead to polynomial systems.
Now. As we know, polynomial functions, in general are famous for approximations. But approximations which can be any faithful to the original funcitons to approximate. And this is true for simple poly funcs like Taylor-series, Poly interpolations, things where clearly our case is multivariate functions... a simple straighforwar gneralization.
Let be that said as a start for the next paragraph.

Linear approximations can sometimes be an option to start going about how to parametrize typical non-linear systems, and I'll show that too ; but the degree of falsification would be, in most cases, too large to manage to capture the colorfullness of the real motion of such systems. That's why we use polynomial systems: because also polynomials are famous for being able to approximate almost any function ; but do allow to reduce at arbitrarily small the degree of falsification... that's why... in one sentence.

But with that said, let's elaborate on that a bit more. By saying that what's the particularity of this story about polynomail dynamical systems.
It's this: that as I intend to provide an external video which then I or someone else will transcribe in wrtten form, about the strainge, some interesting properties of generic non-linear systems, it's claer that the linear systems behave very well, in a numerical simulaiton have excellent nuerical stability and excellent preservation of significant digits for the variables used in a simulator, they are very easily predictable, they are simple and everything, while non linear systems, even if intuitively they are not so bad, they do present some difficulties.
And they can, in some circumsances, turn out to be very bad in the context of a numerical simulation. While the particularit of the polynomial systems is that we chain them in order to do an almost, in the context of a numerical simulation, a practically perfect equivalent of any generic dynamical system.
The fact is that the subtle properties, complications and sometimes bad bahaviour of gneric non-linear systems in the context of a numerical simulaiton of them, fall literally in between the pairs of simulation-steps or our chain of them. The complications fall in between 2 blocks of this chain: and so, they disappear and don't cause any problem for the numerical simulation.

if it helps, try looking at a sequence of parametrized SPLINE or Bezier curve-segments ( the variants with lowest degree of polynomial is fine) as they approximate a generic parametrized curve. If the we plot the curve and it's approximate version on 2 papers, as well as some kind of plot reporting also the the tangent-lines, and the resolution of the media on which we draw the curve or the resolutio of the technique with which e draw the curve falls beyond the amount of difference between the 2 curves, then they are de fact indistiguishable. And to achieve this coincidence of the 2 plots, it's sufficient to make the maximum lenght of the segments enough small: but we do not have to raise the degree of the polynomials to match the curve as well as it's tangent lines. While with a polyline, we could not achive the match of the tangent-lines because we would be too rough, and the by reducing max lenght of each segment woul not even make the diminishing of the difference of the 2 plots decrease gradually. While with polynomials, SPLINE of Bezier curve-segment, yes: because polynomials manage to capture the colorfulness of a curve's nature. And similarly, we approximate a generic system as it's cofiguration evolves from instant to instant, with a polynomial system which we update from stp to step according to some rules, and if the stepping-time difference is sufficiently small, the 2 evolutions of configurations become indistinguishable. And the degree of falsification in the approximation with respect to the original, decreases also gradually ; until it reaches the desired amount. While with linearly approximated parametrizations, this would not happen... it would be far too rough.


I hope I've managed to be sufficiently clear within the small space that a preambule allowed.
So. Let's get started.

How to feed a mechanism's definition into the Simulator's sourcecode? - a very quick outline

So.
As a start, we let for now in the background the precize parametrization rules, which will be discussed in detail in another section. Here, just think to the simple examples shown in part 1/3... : we assume that You have the parametrization of the mechanism whose motion You want to simulate... for example, that You have it on a piece of paper.
And with this said, let me first introduce You to how the parametrization, the definition, the geometry of the mechanism to simulate, is fed into the sourcecode of the Simulator. One You grasped the concept, the detiles abut the parametrization rules to follow in the new flavour of Lagrangian Dyanmics, will fit immediately into the picture.
So. Let's get started.

Consider the quantities we have been writing and discussing in part 1/3 on papaer. The seme quaintities are present in the sourcecode of the Simulator, in specific points, and they musy be assigned the right value according to the mechanism to simulate. In those examples I tired to use the seme names as those used in the sourcecode of the Simulaotr of the correspoinding quantities. Some of which are, with the corresponding name they have in the sourcecode, in the table below.

table 3.1: a simple conversion-table allows to transfer - so to speak - a mechanism's definition,
parametrization, from a hand-written paper-sketch into the C sourcecode of the L.D.N.F.S. Simulator?
__on hand-written paper-examples__ __in the Simulator's sourcecode (in the C programming language)__
N_DOF #define N_DOF corresponding_number_goes_here
N_MPOINTS #define N_MPOINTS corresponding_number_goes_here
xk double pos[3][k]
mk double m[k]
qi double q[i]
... /* discussed later... */
... /* discussed later... */
... /* discussed later... */

Springs and other detailes are discussed later: where also the complete list is given, and illustrated. Thank You for Your patience.

Familiar, isn't it? So... once these get the right values before the Simulator starts to excecute the ideally endlessly repeated simulation-procedure, the only thing remain is the definition of the geometry of the system which is a funciton which assigns the positions of each material point constituting the mecanim to simulate, as a funciton of the q vecotr of generalized coordinates. As simple as that... the function in question, in the sourcecode of the simulator, is called
pos_val( ) ;
and, as You can expect, it uses the float q[N_DOF] ;
variable to get the values of the generalized coordnates,
and throught some custom procedure of assignments, it assigns the right value to each triplet of coordinate in the variable
double x[3][N_DOF] ;
It's pretty much the seme as doin it on paper. And becuase of the story told in the introduction about the chaining of polynomial systems on which the Simulator relies, there's a part in the sourcecode which does that chaining.
it is delimited in the sourcecode of the Simulator as:

//============REOREIENTATION PART================
//.
// all what needs to be done between 2 steps of simulation, goes here
//.
//===============================================

Now, do not worry about this... for linear dynamical systems, that is for systems which are among the simplest, this part of code remains void. When You arrive to the end of this guide, it all shall be claer, and You will be able to, if You want, to put this part of the simulation in an isolated function. I didn't since the chaining part, done by that part of the procedure, is a bit delicate and some may find it disturbing to see implemented as the call to a function whose purpose is unclear.

Hope this quick outline was useful and managed to convey to You, dear Readers, at least some information useful to make You feel comfortable with the Simulator and it's usage... .

How does the Simulator work?

Between 2 folowing steps of simulation, the q and qd values must so as to satisfy the Lagrangian Equatons of Motion, which satates a conservation law as we called it.
Cleary, we know that we don't have to care about q because since in a stepping numerical siulation, it can only do as:
q = q + qd*dt ;
to which we also add this.
qd = qd + qdd*dt ;
that was to expect because the Lag Eq Motion is, rougly, has the form of a 2nd order linear diff eq so it's like stepping to do an animation of an articulated body but without knowing anything or the dynamics of articulated bodies ;
For the dynanics to be realistic, we can't give the qdd term... any value we like between 2 steps... it has to receive the values adequate for making the motion realistic.
We arrived at it: the simulator has to ge the qdd term right, and that means so as that between 2-3 steps, the qd term and through it als the q term satisly the equivalence imposed by the Lagrangian eq of motion.
The correct value of the qdd vector from step to step, comes from the Lagrangian. That said, we are ready for the detailes.


So. Let's get into the details of... how to find the qdd correctly?
The Simulator works simply by solving numerically the system of Lagrangan Equations of motion: which means that each partial derivative figuring in the expression, is calculated as a simple finite difference. For those who are not familiar with the concept of partial derivative, take it like this:
given a function of the form f( q1, q2, q3, ... qN ) and chosen one of it's variables, call it qi, the partial derivative of the function with respect fo qi is the value given by this (scaled) difference:
( f( ...,qi + h,... ) - f( ...,qi,... ) )/h . which is valid as long as h is enough small ; and scaled means that the difference is then divided by h.
And You are fine! So, let's go on!

We rewrite the system of Lagrangian Equations of Moton as
qdd = A-1N_DOFxN_DOF( q, qd ) * b( q, qd )
where some of the partial derivatives of the T( q, qdd ) or L( q, qdd ) function are the various elements of the A matrix and others are summed to make up the elements of the b vector (vector of known terms). So it, de facto, becomes a linear algebraic system:
qdd = A-1N_DOFxN_DOF b ; where, instead of matrix inversion, the simpler Gaussian Elimination algorithm can be used to find the qdd solution-vector.
And then, all that the Simulator has to do, is to excecute over and over again, the procedure based on this, that is:

qdd = A-1N_DOFxN_DOF b

q = q + qd*dt
qd = qd + qdd*dt

As simple as that. And because of the facts and most importantly their consequences outlined before, this is granted to work always fine, that is, in our case to never lead to a division by zero in the procedure to find the qdd values... as division by zero is an illegal operation ; and to have a preservation of significant digits for all variables in play, nearly as good as that which we would have if we simulated a linear dynamical system.

We can here take advantage also of the fact that A is symmetric (as proven in part 2/3 for those who are interested in the theoretical detailes), therefore it's sufficient to calculate the diagoal's values and then to calculate only the values at one side of the diagonal and to replicate the values found to the other half.
But that alone would not be sufficient for a fully automatic numerical simulator, where no formal, symbolic expressions occur except that for the parametrization ( this is not entirely true anyway: some more on this topic later ). We have no symbolic expression for L( q, qd ) ;
and so neither have we symbolic expressions for the components of the matrix which is in the basic procedure of the numerical solution of the system of the Lagrangian Equations of Motion:
(figure, or expression...)
qdd = A-1N_DOFxN_DOF b
So. We simply have to get it by apply the definition of the Kinetic Energy function, or the Lagrangian function. At this point, the concept is the seme for both, but the key to understand the concep is certainly the Kinetic Energy function, as mentioned earlier.
T = 0.5*Sumfrom k = 0 to k = N_MPOINTS( xd2k + yd2k + zd2k ) ;
Where each xdk ,ydk, zdk is a function of the q vector of generalized coordinates.
So far, we can do all our partial numerical derivatives by evaluating the differences beteen different values that the T function yelds by applying the definition.

We are almost there. The only thing that remains to say is how to get to the xdk ,ydk, zdk values, that is the velicity-vector xk of each point constituiting the dynamical system to simulate?
Well, simply by applying the definition.
xdk = Sumfrom i = 1 to i = N_DOF( (d xk / dqi) * (d qi(t) / d t) ) =
= Sumfrom i = 1 to i = N_DOF( (d xk / dqi) * qdi ) ; where we stick to this form

With this, the Simulator would work but would be slow because of the many time the Xk is evaluated... an operation wich would anyway require to avaluate the pos_val() function as many times.
So, in this case, since this, even if this typical symbolic form would suggest to evaluate the xk funcitons really many times, in the Simulator another equivalent is is implemented, to gain en enhancement is speed.
An equivalent which uses a definition with the
a = pos_val() expression, the following way:
ad = ( a (q+qd*h) - a (q) )/h ;

that's all... the pos_val function is evaluated, that is, the corresponding procedure in the simulator is excecuted, only twice. So the Simulaotor works and it is also does so in an optimal way!

If You are not interested in detailes, here it's over. The part on the parametrization rules will explain how, at the end, the simulation procedure so far outlined performs that cheining operation to pass from one polynomial system to another one compatible with the previous one, to do generic simulations insted of only systems exhibiting mid-oscillatins only.
The previus equivalence, which is literally the jolly of the Simulator to be fast, lightweight, intuitively comest from the fact that the material points caomposing the mechanism to simulate, can't exit form the set of - so to say - legal set of posiitons ; and moreover, for a very little eleapse of time we can say tha even if their mosion is ususally heavily constrained, their motion is still in accordance with the simlple principle of inertia, so that if at one instnat the position of a point was
xk( q( t0 ) ),
an instant later it's position is
xk( q( t0 + qd*dt ) ) ; where dt is an elapse of time, as small as possible.
Reason why one can apply equally well the forward-way as well as the backwad-way definition of derivate... either in infintesimal fashion or finite-difference fashion as well.

With this said, for those who are interested in a detailed deduction and proof of all that was said so far about the Simulator's functioning, below is a section dedicated to the detailes.




detailes:
For those a bit familiar with the concept of differential equation and systems of them ( clearly, we consider systems of a set of diff eqs which are all intercompatible among themselves (!!) ), the Lagrangian Equation of motion or system of equations of motion, are one or r systems of differential equations, which se wbring in this form which let's it see in a form from which it's extremely easy to forsee a numerical solution of it:
qdd(t) = A-1N_DOFxN_DOF( q(t), qd(t) ) * b( q(t), qd(t) )
Well, this is very friendly in the context of Numerical Methods, since at first glance has the form of a simple 2nd order linear differential equation, or a system of 2nd order system of linear differential equations, in which the function q(t) is the solution-function to look for, or the q(t) vector of solution-funcitons.
Hence, for those evne just a little bit familiar with such things, a simple stepping numerical solution of the king presented in the previous paragraph is fine, or Ruge-Kutta 2nd order, 3rd or 4th order variants can be used... or Implicit Euler method, or whicever You prefer.
It's extremely simple so far because all the elements in the A matrix and all the elements in the b vector of known terms are turn into constant, whose value changes at each simulation-step. So the complication of the partial derivateives leading to the values turn into constant varying from simulation-step to simulation-step. Fantastic.
But, in order to evaluate the T of L function and its finite-difference partial erivatives with different q and qd values, we need, along the procedure, to evaluate all the velocity-vectors of each material point making up our mechanism to simulate.
Doing the finite-difference time-derivative of each singel Xk point, would be extrmely slow: since the pos_val() is one function which, however, gives, as a funtion of the q vector of gneralized coordinate, the positions of ALL points... although formally the, xk functions are totally indipendent form each other because it I say X2 = X1 + [...], as soon as X1's eapression is sobstituted entirely, all the dpendence of X2 on X1 disappeares. They are de-facto independent, so: pos_val() is a formal function of (q) only, even if it assigens siulataneously the position to all N_MPOINTS material points.
But in a numerical simulator, we dont' have the luxury of basig simultaneous assignments on the presence of a fully developed formal expression for each Xk( q ) point!
Even because, the pos_val( q ) function, in the extreme case, may even rely on bisection and other procedural searches for finding the position of some points, let alone do anything like a formal sobstitution of an analytic expression... . So. Let's go on elaborating a bit on this.

The starting idea is that by getting at the heart of the definition, we can also arrive at a simple but essentially important and practical optimization, to consider all the Xk points in one hit. Since evaluating pos_val() is computationally heavy in the utmost of the cases. So... let's see it a bit.

We could easily apply the numerical finite-difference version of the expression for each xdk :



to use this last one.

But this would require to evaluate lots of times the pos_val() function, since with no explicit formal expression for each material-point's poisition xk as a function of the vecotr of generalized coordinates, we would have to evaluate almost at each time the positions of al points... a computational overkill. Here, it's appropriate to re-examine the definition in its finite-diff method to adapt it to the current scenario to avoid wastes of this kind.

So. We have now gotten at the concept of the pos_val( ) funciton of the simulator. Formally, it would be a vector-valued fuction which, given the q vector, gives a set of 3*N_MPOINTS values, which are all the 3 coordinates of each point of the system, put in a vector. Something like this:

some_vector_with_many_elements = pos_val( q ) ; one shot, one kill.

To say it with the most popular terminology these times' world, the pos_val( q ) function can be considered an explicit (!!) formal function whose output goes into the R3*N_MPOINTS set ; while it's input is in the RN_DOF set. Since values in output are completely independent from each other as intuitive, we only have to restate the definition of the velocity of each point.
[ x1 x2 ... xN_MPOINTS ]

The idea is that so, to find the velocity-vectors of each material point, is sufficient to evaluate the pos_val() fuction only twice (!) .

The fundamental and the only optimization-choice in the Simulator: since as evident, no others are needed at all. So, let's see how this only one is done.

Let's introduce some notation, and then quickly carry out an illustration of the idea. The fundamental notation is this:
[FIGURE: fast velocity eval, line 1]

The symbolic definition for the velocity-vector of each point is:
[FIGURE: fast velocity eval, line 2]
;
where we stick to this form for now, even if we have to keep in mind that since we are in a context in which the derivative of the xk vector is done with finite differences, it may be better to use the original definition of derivative.

da( t ) / dt = ( a( t + h ) - a( t ) )/h ; definition of (univariate) derivative... where left-way of right-way variant (this is the right-way) yelds the seme quantity.
da( q( t ) ) / dt = ( a( q(t+h) ) - a( q(t) ) )/h ; definition of (univariate) derivative, BUT with the explicit dependence of a on q which at it's turn depends on t (depth-2 univariate composite function ), which is equal to
( a( q(t) + qd*h ) - a( q(t)) )/h ; the form which we use for numerical deriation of depth-2 conmposite function in our Simulator... let's call it rough form... where left-way or right-way derivative is the seme.

We do not report the full deduction of it here. However, the basic idea is to use the fact that near to a vlaue t0 at which we evaluate the q(t) function,
q(t0+t) = q(t0) + qd(t0)*t ,
and that thus
q(t0+h) = q(t0) + qd(t0)*h ;
allows us to replace the q(t+h) function in
( a( q(t+h) ) - a( q(t) ) )/h
with:
q(t0) + qd(t0)*h .
Equivalent to:
q(t) + qd(t)*h
by turning the constant t0 into a variable t. To finally yeld:
da( q( t ) ) / dt = ( a( q(t) + qd*h ) - a( q(t) ) )/h .
End.

A fast, one-breath outline of the deduction follows.
We start out with the notation telling what we are trying to do.

Then we write the definition of derivative with the respect to the innerest variable t, for the above-type of function.

We rewrite it with the common dotted qi notation.

It's a total derivative, and so there is a nice big summatory.
But since the summatory is not too clear for us in this case, we rewrite the seme thing with the notation with uses the nabla operator
.

Then we rewrite each element of the nable vector with the raugh definition of derivative as the limit of an incremental ratio, as h tends to zero.

And now, using the concept that with a derivative we can linearly approximate a function around a specific value of it.
And since it's so linear, we can bring the qdi multiplyer inside, as in linear functions of the form considered until now, k*[f(x0) + fd(X0)*x ] = [ f(x0) + fd(x0)*k*x ].

End.

For the mathematically inclined, I've prepared a more thorough proof ( extra reference item 3.1 ) of the general case of the derivation of the composite function such as
a( b( c( e(t) ) ) ) ; but do not get confused by it, since all the components of our vector are only depth-2, so the general rule is NOT required and would only mislead us.

extra reference item 3.1 -NEED TO CORRECT TYPE-MISTAKES HERE, IT'S PRETTY FULL WITH THEM!! -: outline of proof of univariate derivative of generic depth-N single-valued function (for vector-valued it's the seme anyway except that the most external function, in this case "a", is vector-valued.
As a safe start, we take the plain definition of derivative with h tendent to 0,
d a( q(t) ) / dt = ( a( q(t+h) ) - a( q(t) ) )/h ;
were we use the equivalence q(t0+h) = q(t0) + qd(t0)*h, in which we can turn the t0 constant into a generic variable t, to obtain:
q(t+h) = q(t) + qd(t)*h ;
We then sobstitute q(t+h) wih that expression, and thus obtain the equivalence
( a( q(t+h) ) - a( q(t) ) )/h = ( a( q(t) + qd(t)*h ) - a( q(t) ) )/h ; end.
Clearly, in a numerical simulation which uses this fact, we only have values of q and qd, so the expression becomes one reduced to be true to a sinlge step of simulation, or more in heneral a single step of numerical solution, according to:
( ai+1 - ai )/h = circa...
( ai - ai-1 )/h = (( ai + qdi*h ) - ai )/h ; where clearly the ai terms in the last expression delete each other, to leave the rough definition valid in the context of a numerical soluton, which is, I tried to oultine so far, not any far from the analytical context.
expression where i indicates he number of step at which the repeated proceudre has arrived so far.

Which is valid (for those who are inerested in it) because of the seme principle of linearity of the function f(t) near a specific value of the generic variable t the f(t) function depends on, according to:
f( t0+t) = f( t0 ) + [d f( t )/ dt]t0*t .

Now, at this point the idea is simly that of reducing the task of deriving a heavily composite funciton of the form a(b(c(d(t)))) to repeated excecution of the simple, non-composite function's derivative with respect to it's simple variable, as a(b), which we are able to derive.
Now, by using the
f( t0+t) = f( t0 ) + [d f( t )/ dt]t0*t rule,
by doing some sobstitutions, we are able to arrive at the defnition of chain-rule for the derivation of composite functions.
In the generic definition of derivative, ( a( b(c(d( t0 + h ) ))) - a(b(c(d( t0 )))) )/h
, we sobstitute, in a( b(c(d( t0 + h ) ))), the d function with it's linear approximation near t ; to obtain
d( t0+t) = d( t0 ) + [d d( t )/ dt]t0*h , and then do it also for the c function considere as formal a function of d (don't get confused!!)
c( t0+t) = c( d0 ) + [d c( d )/ dd]d0*h , and then do it also for the b function considere as formal a function of b (don't get confused!!)
b( t0+t) = b( c0 ) + [d b( c )/ dc]c0*h , and finally arrive at the top-level function
a( t0+t) = a( b0 ) + [d a( b )/ db]b0*h , And arrive at a modified definition of derivative:
[PUT THE CORRECT FORMULA HERE!!] ( a( b(c(d(t0))) + dbc( t0 )*dcd( t0 )*ddt( t0 )*h ) ) - a(b(c(d( t0 )))) )/h ;
which anyway is sufficient to interpret, trying to guess what that rescaling of h produces in the final value of the derivative. We are almost there! The linear approx of a func near a value, does the job. By applying the definition of linearity, we understand that rescalig h rescales the function's derivative's value! Now very very special care must be done with the equivalence we arrive at:
( a( b(c(d(t0))) + dbc( t0 )*dcd( t0 )*ddt( t0 )*h ) ) - a(b(c(d( t0 )))) )/h = dbc( t0 )*dcd( t0 )*ddt( t0 )*[ ( a(t0 + h) - a(b(c(d(t0)))) )/h ] ;
and the usage of the linearity of the generic f(t) function near a given t0 value... this may seem weird but this is actually a hyptheses which simply underlies the whole calculus! And this equivalence's left or right flavour may be used with greater success in some fields than in others. In numerical derivation, the left may be actually prove to be better too.
This all leads to the traditional Chain Rule because thus it's true that
( a(b(c( d( t0 + ddt(t0 )*h )))) - a(b(c(d( t0 )))) )/h = dd( t0 )*[ ( a(b(c(t0 + h ))) - a(b(c(d(t0)))) )/h ]
with h small enough, the smaller, the better. Then, to arrive at the classical Chain Rule, it's sufficient ot go on with the seme sobstituion with the c(d) function, then with b(c), and finally a(b) ; completely procedural, to obtain finally what the Chain Rule states.
( a(b(c( t0 + dcd(t0 )*h ))) - a(b(c(d( t0 )))) )/h = dcd( t0 )*ddt( t0 )*[ ( a( b(t0 + h ) ) - a(b(c(d(t0)))) )/h ] ;
and so on, until arriving at:
( a( b(c(d( t0 + h ) ))) - a(b(c(d( t0 )))) )/h = dbc( t0 )*dcd( t0 )*ddt( t0 )*[ ( a(t0 + h) - a(b(c(d(t0)))) )/h ] ;
where the derivative-definition at the right is finally replaced with
dab: that is, a's derivative with respect to b.
and finally with it's formal eqauivalent d a(t) dt ; and we have the Chain Rule. end.


All we have in fact, is a buch of funtions of the form
a( q(t) ) ; a depht-2 composite function (!!) .
That's why the intuitive justification provided at in the previus part is perfectly fine as far as Analytical Mechanics and the functioning and use of our Simulator is concerned.


Then we simply try to state a similar explicit and rough definition considering that xk(t) is a multivariate and also composite funciton in this excact way:
xk(q(t))
reason why the definition becomes:
d xk( q( t ) ) / dt = ( xk( q(t+h) ) - xk( q(t) ) )/h ;
it's easy to understand intuitively if we condsider that the xk(q) point simply can't exit from it's allowed set of positions, so left and right derivative must give the seme result, so that we prefer te right-way derivative.
And then we take advantage of these equivalences (which in the symbolic fashion are usually not used but do hold true anyway), and pass to the previously outlined vecotr notation intead of using the definition for a single
Xk:
we denote
[ x1 x2 ... xN_MPOINTS ]
as
a - which can thus be considered either as a 3xN_MPOINTS matrix ; but also a a 3*N_MPOINTS vector of the form
[ x1 y1 z1 x2 y2 z2 ... xN_MPOINTS_x yN_MPOINTS zN_MPOINTS ]
... whicever You prefer.
The important thing in fact is that it can be used as, and behaves as:
a = pos_val()
... that is, as a simple vector-valued function. Function which depends a vector of N_DOF variables. To give a simple
RN_DOF ----> R3*N_MPOINTS
function.


And so finally we write our final form:
They are these:
da/ dt = nabla(a)*qd = ( a (q+qd*h) - a (q) )/h
And since the mid-term is not of much use for us in our context of numerical simulations, we keep the equivalence between the first and last term as our jolly for getting the velocity-vector of each point in a computationally efficient way ; that is , with as few evaluations of the pos_val(...) function as possible:
Now. The equivalenve between, the first and third term,
da/ dt = ( a(q+qd*h) - a(q) )/h
comes from the more-less brute-force manipulation of the definition of univariate derivatie of a composite function in the form
a( q(t) ) ; which is a depth-2 composite function... that's the concept.
For the mathematically inclined, I gave a quick proof ot this case... from which the vector- or matrix-valued case follows immediately, according to the previously written definitions. However, I did not give the proof of the multivariate case. In one sentence, it consists in the generalization to a multivariate function through the concept of directional derivative: very simple. Probably in a later edition of this quide I will outline the complete proof since most people are familiar with the applicatin of the Chain Rule but not with the equivalent of it which we employ.
Anyway, let'go on with out deductions to arrive at the equivalence use in the Simulator: an equivalenence which is very optimal in the numerical fashon when the time-derivative of the xk position-vecotors is to be calculated.

Well, we just have to get to the more general case when the a function depends on more variables, each of them functions of the one "t" variable, this way:
a( q1(t), q2(t), ..., qN_MPOINTS(t) ) ; written more compacly as
a( q(t) )

At this point, consider that a is not just a generic funciton, but it's one of the 3 coordinates of one of the material points of the mechanism whose motion we want to simulate with our Simulator... like this:

| x1( q(t) ) |
| y1( q(t) ) | = x1(t)
| z1( q(t) ) |

| x2( q(t) ) |
| y2( q(t) ) | = x2(t)
| z2( q(t) ) |

...
...
...

| xN_MPOINTS( q(t) ) |
| yN_MPOINTS( q(t) ) | = xN_MPOINTS(t)
| zN_MPOINTS( q(t) ) |


Now we can drag towards some generalization: since (from the point of view of of derivation and similar mathematical operations) the single elements of this vector all have the seme properties of the a( q(t) ) function that we brought up as example, so all the things that hold true for a( q(t) ), hold true for each element of the previously written vector. That is, they hold true for all the vector in its interity.
Once that's set down, it's immediate to see that it holds the seme for the case in which a is a vector of 3*N_MPOINTS elements like the above-written vector, let us call it then the a vector, as well as in the case when a is a 3xN_MPOINTS matrix, which we thus could call a3xN_MPOINTS.
So that we come to the previously stated equivalence feauring
a( q(t) ) or
a3xN_MPOINTS( q(t) )
.

I now will outline a proof of this for those who are interested in it. However, read the proof of the previous pragraph too, where the proof is based on considerations of the properties of mechanisma as we describe them with a prametrization and their respective gneralized coordinates giving the position of each material point composing the system, as a funcion of the vector of genralized coorinates q.
We have the composite funtion
a( q(t) ) ; a depht-2 one.
We write down the defintion of univariate drivateive for it:
d a( q(t) ) / dt = ( a( q(t+h) ) - a( q(t) ) ) / h ; this is not of much use, but let's take a basic property of funcitons and their derivatives, to try to transform this into a different but mathematically and numerically equivalent form.

The equivalence on which I've been elaborating a bit now, may seem strainge to some, and therefore may have a hard time at surviving without at least an outline of a proof: for this, I've prepared a graphical interetation of the proof, relying just on some basic notion of modern multidimensional gemetry so much in harmony with a modern industrial world in whih out Simulator was deviced (fig.).

[INSERT FIGURE]

Enjoy!

As a last note about the previouly discussed equivalences: since some of these equivalences are not reported on most books on either Analytical Mechanics nor mathematics alone, be very careful to grasp them... .
da/ dt = ( p_xk (q+qd*h) - p_xk (q) )/h ;
that's all. Very very easy to implement in any modern programmng language, which in the case of our simulator, is the C programmng language. Have a look at it's sourcecode to see this implemented.

So, we can get in a computationally efficient way the velocity-vector of each material point of the mechanism to simulate, and the simulation-procedure has all to evaluate numerically all the partial derivatives and their sums when needed, to arrive at each coponent of the A matrix and to get to the next step of simulation. That's all. So simple, so efficient.


The last topic, not really optimization but an essential way to get the simulation-procedure to be as close to ideal as possible, also a measure is needed to avoid problems which could derive from the fact that each variable's value is stored with a limited numbr of significant digits.
Clarly, if we would parametrize a system which overall can fit, say, in a box of 10x10x10 meters, then, if we would translate it to kilometers aways from the origin of the global reference frame, we would waste so many significant digits, that the simulator would produce unprecize results, and would easily crash because of the compromission of numerical stability. We mus counteract this.

First thing is to just keep track of how that 10x10x10 (example) box went far from the origin, and just draw the conventional geometrix center of the system at that position, but use
[0 0 0]
values for that position in the simulation-procedure, except that for the contract-check with the ground.
That's simple. Look at what the delta_xc, delta_yc, delta_zc values do in the sourcecode of some of the sample-simulators.

This is really not te stability theorem, but just a wild idea of it. The real stability theorem assumes this precautions are taken, as a hypotheses. The theorem instead, prescrbes in fact, also another rule:
to center the defnition of bodies, even if this cmplicates a bit the praremtrization in most cases. Let the explanatin be done by a simple figure, below.
;
In the part dedcated to parametrization, You will learn quickly how to center bodies even in the less comfortable situation illustrated in the right side of the figure.

Parametrization: rules and techniques - How to define an articulated mechanism for the Simulator?

For those who have already had some exposure to Analytical Mechanics, I mean, in this case to Lagraingian Dynamics, and so to various parametrization techiques and concept underlaying it, I will underline a bit, I'll hint a bit at, the differences and similarities between the rules of the Lagrangian Dynamics' new flavour and it's traditional flavour.
If You haven't had any exposure to these things, then, better! - just concentrate on the parametrization rules of the Lagrangian Dynamics' new flavour's, which are simple and straightforward, and are thus even easier to undestand than the tradition techniques - so to say - of parametrization.


So, from time to time I'll hint at the differences, as reguards parametrization, between the Ladgranguan Dynamics' new flavour, and it's traditional flavour. So, don't worry about getting cofused: it will be all fine.
I say this Because It's important not to get cofused between the 2 flavours, since in using the Simulator, the rules of the Lagrangian Dynamics' new flavour have must be used ; even if this clearly does not limit the possibilities. For now take it so: advanced users can, eventually, progress to using a mixture of the 2 flavours so don't worry.
In the classical approach, yes... paramerization is more a sort of art a question of taste, than a science. In fact we have planty of freedom in the way in which we parametrize a system, as long as the final degrees of freedom, the final motion which a point, or a set of points, or a singe rigid-body or a set of more rigid-bodies, can have. As long as the motions it can do are those intended, it's fine.

While instead in the new flavour of LD used in the SImulator, parametrization is a fixed sceme with rather strict rules. This, to some extent, however simplifies also the job.

In the traditional approach to Lagrangian Dynamics, we are taught to subtract degrees of freedom. From the total number of degrees of freedom which a set of points have, from the total freedoms of motions - so to speak - , which in 2D is the total number of points multiplied by 2, while in 3D it's the total number of points multiplied by 3. And in the case of rigid-bodies, well, the tottal number of DOFs of a set of N rigid-bodies, in 2D, is 3*N ;
and for a set of N free rigid-boides, in 3D, we have 6 DOFs for each, so the total n of DOFs of such a system, is N*6.

And as reguards the joint-mechanisms, in the classical approach we are tatught that these so-called joint-mechansims do subtract DOFs from these total numbers of DOFs I've listed before according to the various 2D and 3D scenarios.
And of course we say that at each constraint, as it is called in the classical approach, a DOF, 1 DOF is subtracted to the total ; and if we consider all the constraints, we have the orignal free systems's number of DOFs, MINUS the number of constraints... so arriving at the total number of DOFs left.
Listen to how it's a bit awkward to say it.
And moreover, different joint-mechanisms, like those I've hinted at before like those listed in the table below, [TABLE for 3D and 2D cases]
[PRISMATIC joint, the Universal JOINT in 3D ]
each does subtract a different number of degrees of freedom, since some of them concretize more than one (so-called) constraint at a time. For completeness, I report each joint-mechanicm, in the cassical approach, how many DOFs subttact: look at the following table below...
and then forget it as soon as You can.
The heart of the question is that in the classical approach, by definition, each constraint subtracts one single degree of freedom. Stop.


Instead, in the new flavour of Lagrangian Dynamics, we ADD degrees of freedom ; starting from zero : very simple!
We start from a rigid-body dince for convenience we restricte to the use of rigid-bodies only - as introduced a while before (part 1/3 of this presentation and guide ), and start to - almost literally - to "build" our mechanical system by adding the various types and numbers of freedom-of-motion to the system by joining bodies to that initial one, through various joint-mechanisms: which, this way, do ADD degrees of freedom.

A joint-mechanism ADDS degrees of freedom in our case.

We construct a mechanicsm by combining rigid-bodies through various joints ; a body can be either free, or held by a joint-mechanism. And when a to a body a joint-mechanism is attached, that body by definition has 0 (zero) DOFs : that's why in our context we say that a joint ADDS, actually, DOFs.


NOTE: it's not only to assure polynomial systems as said before (in the preambule and more in detail in part 2/3), but to make it harder to make mistakes, and make it easier to diagnose mistakes. However keep in mind that these conventions about DOFs and Parametrization apply fully ONLY in the case of systems with NO CLOSED "CHAINS" of BODIES - that is hinted at here, but discussed in other resources than this. Even because they are not common in practical situations axcept te piston mechanism, which anyway does classify only partly as a closed-chain of bodies and thus can be simulated with ease, after consulting resources I will make availale on advanced parametrization techniques. Generic closed-chain boides can be simulated with Lagrangian Mechanics and well as its new flavour, but with some restrictions - discussed in the proper places ; anyways these are very rare and particular applications, so they deserve only the place left to them.

So. Let's discuss how to construct a system, and arrive, in the meanwhile, to the parametrization.
In this book, only the 4 basic joint-mechanisms will be discussed, plus obviously the free body. Look at the tables below!

table 3.2: 3D joints and number of DOFs they add in articulated mechanism restricted to ones made of open-chains of bodies. Exceptions are discussed later.
WARNING: this is a table true in the context of Lagrangian Dynamics' New Flavour... for the classial flavour, plaese look the table dedicated to that.
joint-type DOFs added
PRISMATIC +1 DOF
adding a FREE body +6 DOF
UNIVERSAL +3 DOF
HINGE +1 DOF
SPHERICAL +2 DOF


table 3.3: 2D joints and number of DOFs they add in articulated mechanism restricted to ones made of open-chains of bodies. Exceptions are discussed later.
WARNING: this is a table true in the context of Lagrangian Dynamics' New Flavour... for the classial flavour, plaese look the table dedicated to that.
joint-type DOFs added
PRISMATIC +1 DOF
adding a FREE body +3 DOF
UNIVERSAL +1 DOF


For some additional detailes, read the extra reference item 3.2 ; otherwise, go comfortably on! - You can return to it whenever You want.

extra reference item 3.2 (*1) : detailes on exceptions not discussed here, even because much less typical and musch less frequent in practical applications
Other ones, present but anyway much less frequenty in practical mechanisms, will be discussed in an appendix or straight another book - either in form of video later transcribed to text, or directly as written material.
Examples of such particular mechanisms are (look at the figure):
[FIGURE: a) point tied to 3D curve ; b) point tied to 3D surface ; c) a void cylinder allowed to roll on a surface, with inside it another similar cylinder with smaller radius, obviously, than the containing one's radius ; d) 2D carriege with 2 wheels rolling on a plain surface attached to main body through 2 prismatic joints and a properly positioned spring for each, connecting proper pairs of points of the system ; e) typical 2D or 3D piston ]

Where some of these systems are special inasmuch their simulation requires at least creative usage, and most often also a little customization, modification, of the simulator ; and requires sometimes to understant the limits whithin which the accomplished simulation is valid: a detailed discussion and understaning of my first theoresm on differential equations is vital there: in fact some of these systems are heavily chaotic, but as proved by that theorem, there are 2 types of so-called chaotic systems:
those which, even though under no restriction to polynomial parametrization (old flavour of Lagrangian Dynamics) don't ever cause the simulation to fail at a certain point ( in this case via division by zero, yelding as result a so-called nan in most programming languages) but where the reliability of te result has to be double-checked because slightly different timesteps can lead to a heavy divergence of the result at some point of the simulation ; and those which cause failure anyway in some situations ( even though the restriction to simulation via chaining of polynomial systems), yealding in that specific case to either the attempt to evaluate the square-root of a negative number (yelding nan in most programming languages), etiher the failure of a procedure performing bisection, returning error (that is, the case when no solution is found). All these are hevily non-linear systems, and analytical solutions are not even feasible in the second case, while in the first, I really don't know, but at least they are well-defined: as to these, if You do know if there is analytical solution, better... if not, don't worry anyway as it would have little practical use and certainly it would be rather complicated.


So. Let's start from the prismatic joint: that's the seme as in the traditional approach, so it's all fine unil now.

1.) the PRISMATIC joint (in 3D and 2D: it's well defined in both cases)

[YET TO WRITE or record in video and than to trascribe...]


And now let' come to the parametrization-rule for this, and some further deepening.
Essentially, it's
xk = S1 + q*d ;
where d may be unit-lengt or also not unit-lenght. I suggest to keep it always unit_lenght anyway. Combining more prismatic-joints is as follows:
A prismatic joint needs a body to which to attach it except if it is attached to a fixed point. So, once we have a body attached with a prismatic joint to a fixec point, we can attach another bod to that body through a prismatic joint attached to that body, to which so we attach another body through a prismatic joint.
Clearly, we have then S2 not fixed, but seme as the body1's canter plus an arbitrary vector of decentering, PROJECTED ONTO THE BODIES' REF-FRAME (!!!).

So, assuming that x2 belongs to body 1 and x5 to body 2 at the point x2, the parametrization would be:

x2 = S1 + qi*d1 ;
x5 = x2 + S2 + qi+1*d2 ;

where the indexes on d are clearly arbitrary, as long as one is being coherent. Clearly, more bodies can be attached to body 1 through more prismatic joints. It's only a question of crativity, as this rule of combining rimsatic joints is pretty much sraightforward.
Here it's very important to observe that the parametrization rules so far outlined for prismatic joints, don't take into account the orientation of body 2 and assume it has identity orientation... that is to say, an orientation which does not change in time. In the next subsection we will discuss how to attach a prismatic joint to a body whose orientation, can change over time and it thus depends on, say, q0, q1 and q2, while the body 2 is attahced to that body with a prismatic joint displaced along it's original vertical axis, which we call R in this book. It's straightforward to get familiar with the facts to take into account, as soon as the next section becomes clear. As a final note, when one just wants to attach boy 2 in such a scnearo to body 1 but woithout caring about the orientation of body 2, one can assume the orientation of body 2 to be identity... that's like saying it's just a dumb material point. The other, more realistic and concrete cases, are discussed in the next subsession.

comments:

Mechanisms which are made only of primatic joints connecting bodies, are among the simplest: they all lead to linear systems, that is, the simplest possible dynamical systems - a subcase of the more generic polynomial systems on which our simulator and uderlaying theory relies. However, and this is important to note, mechanism made only of bodies connected by prismatic joints are not the only linear, that is very friendly and simple, systems. Also systems containing perfeclty circular motion-trasmission disks and ideal ropes wrapped around them (figure), are linear dynamical systems, in fact. These are important too, but we do not discuss it in detail here, except the simple idea that in general, the wires's total lenght must be enforced. Once this is understood in 2 dimensions, it's immediate to generalize it to the 3D case, in which the different segments of a wire can wrap around also disk which are have different axes of rotation. The idea is the seme also for closed-circuit wires like transmission-chains or transmission-belts in engines. In the tridimensional case, the principle is the seme: to enforce the fact that the total lenght of the wire or belt or rope, is constant. The dynamics and principle of gear-ratios and such thigs, comes out from this by itself. Let examples be discussed in some appendix, since these types of mechanisms are not the primary focus here, as they usually are part of more generic mechanisms whose parametrization I have yet to introduce to You.
And so are systems which are a combination of these 2 cases.

For both situations, as well as for the combined scenario, it's the case to restate our use to classify systems as linear or polynomial, according to the Kinetic Energy function, and not the Lagrangian function.
For the seek of completeness, clearly it's easy to not theat if one wants also the Lagrangian function
L(q, qd) = T(q, qd) - V( q )
to be linear or polynomial, also the potential energy function V(q) has to be compatible with a linear, or polynomial, dynamical system. Cleary, as discussed in part 2/3 of this presentation, in a system which is strictly linear, we have Kinetic Energy function T which in which no q contributes (or we could also say that the contribution of each of them is null), so if we want to be strict, for linar systems we would write:
L(q, qd) = T(qd) - V( q )
and not
L(q, qd) = T(q, qd) - V( q )
.
But since linear dynamical systems are a subcase of polynomial dynamical systems, it's best to consider the most generic form even because linear dynamical systems alone can't make justice to the colorfulness of practical ( as well as theoretical ) needs. So, let's go on to discuss a more advanced case: the free body, to which we can readily attach other through prismatic joints to begin dealing with the wide spectrum of advanced practical mechanisms.

2.) the FREE body... that is, +1 body with no constraints (in 3D and 2D it's well defined in both cases)

In 3D, the position of the conventional goemetric center of the body will be the vector
[ q3 q4 q5 ] ;
that was easy.

While for it's orientation, well, it is my first theorem on numerical methods:

R( q0, q1, q2 ) = Id + E + E2 =

1 0 0
0 1 0
0 0 1
+
0 -q2 q1
q2 0 -q0
-q1 q0 0
+
-q22 -q12 q0*q1 q0*q2
q0*q1 -q22 -q02 q1*q2
q0*q2 q1*q2 -q12 -q02

Like the visualization of matrices with HTML tables? Not bad, eh?
And now, clearly we can say that each point of the body will be expressed by:
xk0+k = [q3 ; q4 ; q5] + R*x0k ;
where clearly, the x0k things are the coordinate-triplets of the body's each point IN THE REFERENCE-FRAME OF THAT BODY... that is, how that body is defined in the standard unitary-orientation cartesian reference frame, that is when the P, Q, R versors are respectively

P = [ 1 ; 0 ; 0 ]
Q = [ 0 ; 1 ; 0 ]
R = [ 0 ; 0 ; 1 ] ;

with the conventional geometric center of the body at the origin, that is, at the position [ 0 ; 0 ; 0 ] .

Now, let's show some figures illustrating what goes on when we do this parametrization of the free-body, in practice.
[INSERT FIGURES]

As promized, we have now entered the regime of advanced mechanisms. The next paragraph illustrates at least one example of such mechanisms.


examples:
Well, we can now constuct a car or a tricycle, or a vehicle with any number of wheels. By attaching wheels to a main body, through primatic joints. For now obviously wheels are just bodies, they can not rotate, but we are there... . We can already define the dynamics of a vehicle slipping on ideal ice, for example. We can then bring this concept further, but let's do it at the end. Where there is the tricycle and quadricycle example implemented for You of such a scneario, with the LDFS Simulator.
So, a tricycle can be modeled as the figure below shows.
[INSERT FIGURE]
;
The theoretical definition in this way clearly assumes that this mechanism is at the origin of our global reference frame, floating in the void... with a generic air-friction applied to each material point constituting the dynamical system, but with no gravity ; plus the possibility to interactively apply a force to each of the points.

As far as that, the condition of mid-oscillations is met. But that operation is discussed in the later subchapter, and if You feels so You can already consider examples which assume the chaining operation between the simulations steps. Read on if You feel so.

Since the previous mechanism is a bit too theoretical and has the restriction to displacements of orientation which stay within the limits of mid-oscillations, clearly does not alow to try out the transportation capabilities of the vehicle. So, as one could expect, the Simulator makes available incorporated, some utilities to allow letting the vehicle down on a terrain or an ideal flat plane and use some basic contact-model typical of wheels. For now we assume the usage of a simplified model as we haven't yet discussed the hinge-joint which instead allowes to define also the freedm of each bodies used as wheels, to rotate around their respective axes.
It may seem unusual that I come out mostly with typical free-roaming examples instead of the classical room-experiment mechanisms, but I do so to avoid comunicating prejudices. As the simulator can do both kinds of mechanisms.

Some trials by modifying the basic examples allows to get quickly comfortable with the simulator's usage. For example, try to modify the free-roamig tricycle example to simulate the tricyle in a gravityless ideal world, in which we construct a sort of vibration-test mechanism by attaching 4 points of the tricycle's main body to 4 fixed points, with 4 springs. Then activate gravity, and try to stimulate the mechanism by applying some forces here and there on it.
But remember to deactivate the ground-contact force, as the origin may be under the local level of ground! While giving up being near origin would not meet the requirement for the maximum preservation of significant digits ; which, in the free-roaming examples is already automatically taken care of, but which asssumes that ALL the point of the system to simulate are enough close to the
[ delta_xc, delta_yc, delta_zc ]
position.
Then, when You have read the next 3 paragraphs instead try to to simulate also, for example, a scenaro in which the tricycle is hung to a fixed point near the origin, taken by one of it's points, through a universal joint ;
and then a spherical one, for example. And then activate gravity to see how it moves. Just be careful that the universal joint is a bit unstable, so it may be necessary to activate the RK2 more-precize numerical method. Of Course if You uncomment some lines of code a that part, You can well upgrade it to RK3 and RK4 as well. Even if these are by far too slow for real-time simulations, so are not suitable for games or real-time simulations for example. While the spherical joint has a good numerical stability as well as the hinge-joint, so with that it's fine.

3.) the UNIVERSAL joint (in 3D and 2D it's well defined in both cases)

Well, the universal-joint is essentially a variation to the free body. But it's all about it's orientation being free while the position of the point at which it is supposed to be hung up to a fixed point through a universal joint, is fixed.
The following sequence of fiugres will illustrate roughly the concept.
[FIGUREs: universal joint]
;
comments: ...

;
comments: ... .

Once this concept is there, it's just sufficient to elaborate on this until the meaning of what that point is, becomes clear: so that then we can define it comfortably according to any need. [...]


examples:
A very simple classical room-experiment mechanism which can be done with the universal-joint, is the 3D double-pendulum made of 2 rigid-bodies constructed with universal-joints. Look at the figure below!
[FIGURE: 3D double-pendulum made of 2 rigid-bodies constructed with universal-joints... it's a sort of physical double-pendulum, of full right ]
;
In this case the axes are the default ones.
For this example, as long as one does not cause oscillations of it beyond the limits of mid-oscillations, no notion is required on the chaining operation between the simulation steps which are supposed to stay within the limits of mid-oscillations, so this is good. If one onserts the chaning part in the simulation, the mechanism can explore all the freedoms of motion which are possible with the spherical joint. Among the sample-programs implemented with the LDNFS Simulator, this second, generic case is implemented.

4.) the HINGE joint ( defined in 3D only )

This may seem the trickiest to some. So let's take the adequate space to discuss it. Because there's a particular case and a general case too ; even if the general case may seem easy to some to grasp as the particualar case may instead result a bit confusing.

Let's start from the simplest case: a body hung up to a fixed point with a hinge-joint whose axis is defined by a d0
versor. Actually, as You could excpect, since allover the simulator we have generalized coordinates, it needs not be unit-lenght, that is, it needs not be a versor. But using a versor is better anyway.
Well, we must let the position of the points of the body be rotated around that versor, starting from unitary orientation. Orientation, yes it is.
As soon as it's clear that we have to just eaborate on the orienation, we can be sure that it's something similar to the hinge-joint. but We add only 1 DOF and require the change of orientation be like a rotation around ours d0
versor.

Well, a wild guess could be to apply the definition of what the oriantation matrix is, and say that well, I fix the direction
d0
which ideally would perform a unit-angle rotation of the points arount that very axis, and then try to see what happenf if I modulate it's lenght but not it's direction:
0.01*d0 ;
does it make sense? Yes! It will produce an R matrix which rotates the points around the d axis, by an angle of 0.01 radians.

Well, we just let that wild 0.01 amount to be a new degree of freedom to the body's points, like q2*d0
and it's there!
We construct the R matrix for the universal-joint from that
q2*d0
triplet of values, and then choose decentering radius as for the universal-joint, and we have the parametrization of our body with it's freedom to rotate aroung a d axis, by an angle of q2.
It's all here. We don't even have to rewrite the universal-joint's parametrization rules since they are the seme.

The hinge-joint thus, is parametrized as a universal-joint with limited freedom of rotation.
But let's analyze a bit how this is done with more bodies, since the choice of how we define this limitation, must be clear. As for more bodies, the choice of the axis of rotation may not be arbitrary.

Now, the general case: a body_1 attached, with a hinge-joint, yes, but now not to a point of the global reference-frame but to another body, body_0.
We must understand and define well, what axis of rotation to choose... an axis in the global-frame, or an axis in the ref-frame of body_0?

To resolve this doubt, well, just look at how a real hinge-joint is constructed and how then 2 bodies can be joined with it: [figure]
It's the ref of body_0, clearly.
Now, the utmost general case: a chain of bodies, with also bi- or tri- or n-forkations at some bodies... THOUGH NO CLOSED-CHANINS (remember the restrictions mentioned earlier! ).

So: bodyn+1 is connected through a hinge-joint to body bodyn ;
and thus the axis of rotation is a generic
d0
axis DEFINED IN THE REF-FRAME OF
bodyn ;
the illusation below and some examples should do it to make You comfortable with this rule.
.

An important note to winish with is, however to note that the typica wheel-mechanism is also formealized as a body attached to another throught a hinge-joint. The only particularity here is that in this case the body which act like the tyical "wheel", is not only attached at the rotating axis at it's own center-of-mass, but that it's also geometrically symmetric (a fact which is relevant only then we want to simulate also the functionality of a wheel as such when it makes contact with a surface acting as a typical "ground" ).
The figure below tries to illustrate this concept ; while for the rest there are the examples implemented with the LDNFS Simulator.
.
Now let some examples follow, in the next pragraph.

examples:
As the main resource for this, check the pos_val() function of some of the simplest sample-simulators. Among those which are ready at the time I'm writing this, are:
1.) The Tricycle with Pendulum, and the
2.) The Tricycle and Car (quadricycle) modeled with the rotation of the wheels considered.

An important example, to be published soon, instead, is a sort of non-ammortized tricycle ; and a subcase which is passive so not a real vehicle but very important also for the Epilogue of this book, is the non-ammortized, 2-wheelded trailer, whose figure is reported below.
[FIGURE: 2 wheel non-ammortized trailer]

5.) the SPHERICAL joint ( defined in 3D only )

Actually, this is a sort of false case because a spherical jont can be, and in practice it really is, the combination of 2 hinge-joints. So, as the hinge-joint, also this one requires some care. But once we understand it's basic logic, we can parametrize a spherical-joint in one hit, so shortening parametrization and making less likely that we commit errors while doing it. So, let's get started!

I won't be detaled here since this is not so crucial as a joint-mechanism, but I try to do well anyway. as for the hinge-joint we needed a full ref-fare of a body, so will we need one with the spherical joint except it's attached to the global reference frame, to a fixed point.
We try to look a the way we used the R matrix in the case of the hinge-jont. We see that we could vary this a bit... some examples follow

[ u0 u1 u2 ] = q*[ Px Py Pz ]
[ u0 u1 u2 ] = q*[ 1 0 0 ]

[ u0 u1 u2 ] = [ q0 q1 0 ] = q0*[ 1 0 0 ] + q1*[ 0 1 0 ] ; promittent but it's referred to the global ref-frame... well, if that's the purspose, it's fine so.
[ u0 u1 u2 ] = [ q0 q1 q2 ] ; alt! - it's the free-body's orientation!! not spherical joint, does not meet the required numbers of DOFs .

[ u0 u1 u2 ] = q0*[ Px Py Pz ] + q1*[ Qx Qy Qz ] ; this promizes... but is the combination legal? Yes! It's this.


Well, the last formula is it! Meditate on it! You just have to choose between 3 cases:

[ u0 u1 u2 ] = q0*[ Px Py Pz ] + q1*[ Qx Qy Qz ] ;
[ u0 u1 u2 ] = q0*[ Px Py Pz ] + q1*[ Rx Ry Rz ] ;
[ u0 u1 u2 ] = q0*[ Qx Qy Qz ] + q1*[ Rx Ry Rz ] ;

So easy. It is the mathematical combinatin of 2 hinge-joint parametrizations. Just be careful to which body to tie another one with a spherical joint, and careful to the coice of the axis-pair.

Once we geve this warning, a very simple classical room-experiment mechanism which can be done with spherical joint is the spherical double-pendulum. Look at the figure below!
;
In this case the axes are the default ones.
For this example, as long as one does not cause oscillations of it beyong the limits of mid-oscillations, no notion is required on the chaining operation between the simulation steps which are supposed to stay within the limits of mid-oscillations, so this is good. If one onserts the chaning part in the simulation, the mechanism can explore all the freedoms of motion which are possible with the spherical joint.
Among the sample-programs implemented with the LDNFS Simulator, this second, generic case implemented.
The spherical joint, however it may seem tricky at first galance, it used for example in the mechanims which gives extra ammortization to the driver's cabine in modern trucks: the cabine-body can move forward with respect to the rest of the basis, and move left and right... bit can can not turn around it's axis defined as the axis which says the "up" direction with respect to the cabine. the implementation of that is left as an excecize: You can start imlementing it by attaching a box to the 4-wheel car example (the 10 DOF one is better for this than the 14 DOF one), through a spherical joint, keeping in mind that the spherical joint must allow rotation with respect to the main car-body's OWN P and Q axes.

The Chaining operation: How to go from mid-oscillatory systems to generic systems? And so, what to put in the reorientation part of the Simulator's sourcecode?

Now. We have already mentioned the possibility to reset the overall 'center' of our articulated mechanism to
[ 0 0 0 ]
after each step to then accumulate the variation in that position into a
[ delta_xcm . . ]
vector which is added ONLY when expressing numerically in a table the positions of each point of the system a the end of each simulation-step, as well as, coherently, in the graphical representation and for the check of contact with ground and such extra things not part of the simulation itself.
This can be done because translation of that center does not influence the T() value and moreover is a linear operation in general, so letting x, y, z vary, or accumulating it in an auxiliry value and then resetting that x,y,z to 0, is the seme thing.
Well, this is certainly a big deal for the numrical simulation as each variable has limited number of significant digits so that a value of x,y, or z of thousands of units can destroy precision ; which so with resetting does not happen.
But let's concentrate on the mathematical validity of this operation. We said why this operation was legal. Fine. This is a very simple kind chaining, but it's not the real one: as x,y, z do not influence T()... and we prohibited to let it compeare in V() as well (!!). In fact alone, it would not suffice. But this is not a book on the theory of polynomial dynamical systems and the class of diff-eqs it leads to, so let's just take this as an application of the stability theorem.
But as a vague concept, it's good for a start. In fact the real thing comes now.


The idea is that also orientation must be chained in a similar way. Let's oultline quickly how this is done and why it is legal.
The conventional geometric center of each body, clearly remains the seme. But once at the end of a step, a total displacement, a total variation in the oriantation of each body, is achieved.
And since the prametrization of the orientation for each of these is polynomial, and of a degree as low as 2, we can do something.

for the system, AT THE BEGIINING OF A STEP, wheter a body has an orientation of
[ -lim < q0 < lim ; -lim < q1 < lim ; -lim < q2 < lim ]
or has an orientation of
[ 0, 0, 0 ] (identity-orientaion) ,
is the seme. As well as the very shape, very dfinition of a body is the seme.

But we have been talking about rotations: and as an isometric transformation (don't need to know thr detailes here), they can be composed... they are on-linear in shape, but they have the particularity to be composable. Composing rotations of doing successive rotation around varous axes, lead to the seme change in orientation of a body!
So... maybe we can forsee some allowed operations which can be useful, is not straight resolutory for us.


Let's not be too rigorous, for the detailes, some intuition is sufficient, and I'll make available some extra clarifications in some appendix.
So... if at the beginning of a simulation step the orienatation of a body was
[ 0 0 0 ] ,
and if at the end of each simulation step we reoriented the definition of that body (in it's ref rame) of the excact [ -lim < q0 < lim ; -lim < q1 < lim ; -lim < q2 < lim ]
amount achieved in that step, and then resetted the concretrized orientation to
[ 0 0 0 ] ,

it would be fine. Let's see the figure of such a hypothetical sequence, below!

;
comment: ...
;
comment: ...
;
comment: ...

Note that the rule that each body must have it's own reference frame
Pn, Qn, Rn (not present in the 2D cases) ,
is met also here, as stated in part 1/3 . So, without too much verbal explanation, just look again at the sequence and not that the body's orientation changes just excactly the seme way as the orientation of the Pn, Qn, Rn (not present in the 2D cases) triplet or versors used as a reference frame.


So, the next thing is to see that we can do this thing collectively for all the bodies present in the mechanism to simulate. Look at the figure below!

;
comment: ...
;
comment: ...
;
comment: ...

It's a 2D example, to let You see easier what happens, in principle.
Just don't get confused by the fact that bodis are attaches, so the overall change of the system may see exagerated... even if it is not... just see each body how little changed with repsect to its own re-frame.
If You are comfortable with te concept, just think this generalized to a 3D articulated body, such as the one illustrated in the figure below.
[FIGURE: GENERIC 3D articulated body]
;


Now, a last clarification.
Although in the prametrization we used the poly form, for small changes in orientation, reorienting and then restarting from id orientation, and not reorienting but restarting from the seme values of the parameters of the orientation as in the previous step, is the smee thing.
As a last note, in the reorientation operation, we can use the excact rotation, since it's not part of the simulation procedure, and so cannot cause failure, and neither does violate any of our hypotheses outlined in the introduction.

That was it. We have chained 2 intercompatible polynomial systems. The datastructure, that is the polnomial terms in the parametrization remain the seme, but some of the parameters are changed and some of the variables are reset to 0 betweeen to numerical diff-eq-sys solution steps.
After all, it's NOT all that difficult!

How to feed a mechanism's definition into the Simulator's sourcecode? - a DETAILED outline

discussed in video (a provvisory one!) ; the video is at this link:
https://youtu.be/oRgBemz-avo

Epilogue

what are linear dynamical systems, where can they be considered approximations of systems to do small-osc analysis?
What is linearization, when can it be done and what is good for when it can be done legally?
And finally... what happens if we apply the chaining operation to a brutally linearized system even when linearization was not really legal?

A linear dynmical system is a subset of the more generic polynomial dynamical system. But since the concept of linearity is mentioned on so meny references, it is importanto to claerify and demistify this concept since we have been dealing until now with polynomical systems.

With linear dynamical systems, the good thing is that even if in few cases systems are linear and in very few cases a polynomial system can be roughly approximated with a linear one, when we have a linear dynamical system, once we have a linear dynamical system, it's system of differential equations is linear... a system of ordinary differential equations. And for this, not only analytical solutions exists, but when we build an intereactive application which relies on a numerical method, we have theorems for each numerical method, which tell the conditions in which the numerical solution is reliable, tell how much, and tells in which cases it will fail (a case called usually divergence of the numerical solution).
Moreover, at the light of the Lagraingian Dynamics' New Flavour, which relias entirely on polynomial systems, a thorough knowledge of linear systems acquires new uses beside thos I've tried to outline so far. They becoem useful because polynomical systems, as all the systems they approximate to arbitrary degree of faightfulness, have all a linear core as outlined in part 2/3: they are all built around, so to speak, a linear core. So, when designing a polunomical system (a common non-linear system), we can use the tactique of drafting a linear prototype, check it's stability, oscillations, own-oscilaltions, and whatnot, choose the parameters for the numerical simulation to be stable and precize, and then we proceed to add all the various 'corrections' to it in the case of free bodies, universal-joint, hinge-joints, spherical joints and special advanced joints not discussed in this book, and then chech it the ultimate design still behave as it's linearized 'draft' design. Moreover, understanding this, can also make us feel on a more solid ground. We will, in fact, come up with a nice set of examples in which the 'linearization' falsifies little, even if it's about quite complex systems. We well so be able also to build mentally also a sort of ABC of the dynamics of articulated bodies. An ABC into which go the stronges and most primitive relations, while the subtle ones are left outside.
And as a final topic, maybe the most fascinating one is this: what happesn if we linaerize a polynomial system even if this is not legal, and we chain the simulations-steps so constucted? Now. I don't pretend to give a full answer to this, but a comparatively lenghy discussion of this, is absolutely necessary, and can well open up new ways in application... as long a one keeps in mind what is being done. In one sentence: if we take any of the sample-simulators choosing one which has an evidently formidable dynamics ( that is, a rahter complex and colorful one ), what happens if we replcee the
Id + E + E2
expression with the raw linear
Id + E
expression?

When does it lead to a truly linear dynamical system, and when not? And if it leads to one, how much does it falisificate the dynamics of the original mechanism? And if it comes out non-linear anyway, is it beacuse it is impossible to formally approximate linearly, or is it possible anyway with some trick? - and if it's possible, how would it be?
Many questions, which also require to answer if it is possble to linearize all articulated mechanisms at whatever degee if falsification, as long as the final simulation is at least believeable.


This epilogue is dedicated to these things.

So. Let's get started.
[...]

This is the end of the book. After this, You can comfortably read any of the appendix sessions, as well as future books I plan to write on the advanced topics mentioned in this book ; as well as consulting any material on the topic with a fair amout of comfort.

Simon Hasur,
2015 November,
Italy


For those who may want to use the matrix-equation authoring techique used in this book with HTML tables, I reposrt a protorypic one below. It's sufficient to copy the HTML sourcecode of this part, modify the table size, and format it with CSS syntax (where You can adopt, as a start the one I used for the matrix expression in the secion on the fre 3D rigid-body's orientation). TEST of FORMULA, and mostly matrix-equation AUTHORING IN HTML PAGES:

a11 a12 a13 a14 a15 a16 a17 a18
a21 a22 a23 a24 a25 a26 a27 a28
a31 a32 a33 a34 a35 a36 a37 a38