## MaplePrimes Commons General Technical Discussions

##### The primary forum for technical discussions.

by: Maple 17

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations.
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

## Announcing Maple 2016

by: Maple 2016

You, I, and others like us, are the beneficiaries of decades of software evolution.

From its genesis as a research project at the University of Waterloo in the early 80s, Maple has continually evolved to meet the challenges of technical computing.

## Use Eigenvalues, not fsolve@Determinant

by: Maple 18

There has been a spate of Questions posted in the past week about computing eigenvalues. Invariably, the Questioners have computed some eigenvalues by applying fsolve to a characteristic polynomial obtained from a floating-point matrix via LinearAlgebra:-Determinant. They are then surprised when various tests show that these eigenvalues are not correct. In the following worksheet, I show that the eigenvalues computed by the fsolve@Determinant method (when applied to a floating-point matrix) are 100% garbage for dense matrices larger than about Digits x Digits. The reason for this is that computing the determinant introduces too much round-off error into the coefficients of the characteristic polynomial. The best way to compute the eigenvalues is to use LinearAlgebra:-Eigenvalues or LinearAlgebra:-Eigenvectors. Furthermore, very accurate results can be obtained without increasing Digits.

 The correct and incorrect ways to compute floating-point eigenvalues Carl Love 2016-Jan-18 restart: Digits:= 15: macro(LA= LinearAlgebra): n:= 2^5:  #Try also 2^3 and 2^4. A:= LA:-RandomMatrix(n): A is an exact matrix of integers; Af is its floating-point counterpart. Af:= Matrix(A, datatype= float[8]): P:= LA:-CharacteristicPolynomial(A, x): P is the exact characteristic polynomial with integer coefficients; Pf is the floating-point characteristic polynomial computed by the determinant method. Pf:= LA:-Determinant(Af - LA:-DiagonalMatrix([x\$n])): RP:= [fsolve(P, complex)]: RP is the list of floating-point eigenvalues computed from the exact polynomial; RPf is the list of eigenvalues computed from Pf. RPf:= [fsolve(Pf, complex)]: RootPlot:= (R::list(complexcons))->      plot(           [Re,Im]~(R), style= point, symbol= cross, symbolsize= 24,           axes= box, color= red, labels= [Re,Im], args[2..]      ) : RootPlot(RP); RootPlot(RPf); We see that the eigenvalues computed from the determinant are completely garbage. The characteristic polynomial might as well have been x^n - a^n for some positive real number a > 1.   Ef is the eigenvalues computed from the floating-point matrix Af using the Eigenvalues command. Ef:= convert(LA:-Eigenvalues(Af), list): RootPlot(Ef, color= blue); We see that this eigenvalue plot is visually indistinguishable from that produced from the exact polynomial. This is even more obvious if I plot them together: plots:-display([RootPlot(Ef, color= blue), RootPlot(RP)]); Indeed, we can compare the two lists of  eigenvalues and show that the maximum difference is exceedingly small.   The following procedure is a novel way of sorting a list of complex numbers so that it can be compared to another list of almost-equal complex numbers. RootSort:= (R::list(complexcons))-> sort(R, key= abs*map2(`@`, signum+2, Re+Im)): max(abs~(RootSort(RP) -~ RootSort(Ef)));

## Maplesoft at Joint Math 2016 in Seattle

by: Maple

The Joint Mathematics Meetings are taking place this week (January 6 – 9) in Seattle, Washington, U.S.A. This will be the 99th annual winter meeting of the Mathematical Association of America (MAA) and the 122nd annual meeting of the American Mathematical Society (AMS).

Maplesoft will be exhibiting at booth #203 as well as in the networking area. Please stop by our booth or the networking area to chat with me and other members of the Maplesoft team, as well as to pick up some free Maplesoft swag or win some prizes.

Given the size of the Joint Math Meetings, it can be challenging to pick which events to attend. Hopefully we can help by suggesting a few Maple-related talks and events:

Maplesoft is hosting a catered reception and presentation ‘Challenges of Modern Education: Bringing Math Instruction Online’ on Thursday, January 7th at 18:00 in the Cedar Room at the Seattle Sheraton. You can find more details and registration information here: www.maplesoft.com/jmm

Another not to miss Maple event is “30 Years of Digitizing Mathematical Knowledge with Maple”, presented by Edgardo Cheb-Terrab, on Thursday, January 7 at 10:00 in Room 603 of the Convention Center.

Here’s a list of Maple-related events and talks:

Wednesday, Jan 6, 10:20, Room 2B, Convention Center

Lina Wu

30 Years of Digitizing Mathematical Knowledge with Maple

Thursday, Jan 7, 10:00, Room 603, Convention Center

Edgardo Cheb-Terrab

MAA Poster Session – Collaborative Research: Maplets for Calculus

Thursday, Jan 7, 14:00, Hall 4F, 4th Floor, Convention Center

Challenges of Modern Education: Bringing Math Instruction Online

Thursday, Jan 7, 18:00, Cedar Room, 2nd Floor, Sheraton Center

Using Maple to Promote Modelling in Differential Equations

Friday, Jan 8, 10:40, Room 617, Convention Center

Patrice G Tiffany; Rosemary C Farley

If you are presenting at Joint Math and would like to advertise your Maple-related talk, please feel free to comment below, or send me a message with your event and I’ll add it to the list above.

See you in Seattle!

Daniel

Maple Product Manager

## Apply to be a Maple Beta Tester!

by: Maple

As the year draws to a close, we start looking forward to a new year and a new release of Maple. With every new release comes many new features and updates to explore.

We are looking for several new beta testers with a good working knowledge of Maple; We need your input, your ideas, and your experience with our products to help us improve the software and get it ready for general release.

There are many benefits to becoming a beta tester:

• You’ll get to use the new software before anyone else does.
• You’ll help us make our software better in ways that work for you.
• Your suggestions could determine the future direction of the software.
• You’ll get feedback right from the development team.

If you are interested in becoming a beta tester for the next version of Maple, please email: beta (at) maplesoft.com for more information.

## Apply to be a Maple Beta Tester!

by: Maple

As the year draws to a close, we start looking forward to a new year and a new release of Maple. With every new release comes many new features and updates to explore.

We are looking for several new beta testers with a good working knowledge of Maple; We need your input, your ideas, and your experience with our products to help us improve the software and get it ready for general release.

There are many benefits to becoming a beta tester:

• You’ll get to use the new software before anyone else does.
• You’ll help us make our software better in ways that work for you.
• Your suggestions could determine the future direction of the software.
• You’ll get feedback right from the development team.

If you are interested in becoming a beta tester for the next version of Maple, please email: beta (at) maplesoft.com for more information.

## Would Brunel Have Used a Spreadsheet?

This is a post that I wrote for the Altair Innovation Intelligence blog.

I have a grudging respect for Victorian engineers. Isambard Kingdom Brunel, for example, designed bridges, steam ships and railway stations with nothing but intellectual flair, hand-calculations and painstakingly crafted schematics. His notebooks are digitally preserved, and make for fascinating reading for anyone with an interest in the history of engineering.

His notebooks have several characteristics.

• Equations are written in natural math notation
• Text and diagrams are freely mixed with calculations
• Calculation flow is clear and well-structured

Hand calculations mix equations, text and diagrams.

Engineers still use paper for quick calculations and analyses, but how would Brunel have calculated the shape of the Clifton Suspension Bridge or the dimensions of its chain links if he worked today?

If computational support is needed, engineers often choose spreadsheets. They’re ubiquitous, and the barrier to entry is low. It’s just too easy to fire-up a spreadsheet and do a few simple design calculations.

Spreadsheets are difficult to debug, validate and extend.

Spreadsheets are great at manipulating tabular data. I use them for tracking expenses and budgeting.

However, the very design of spreadsheets encourages the propagation of errors in equation-oriented engineering calculations

• Results are difficult to validate because equations are hidden and written in programming notation
• You’re often jumping about from one cell to another in a different part of the worksheet, with no clear visual roadmap to signpost the flow of a calculation

For these limitations alone, I doubt if Brunel would have used a spreadsheet.

Technology has now evolved to the point where an engineer can reproduce the design metaphor of Brunel’s paper notebooks in software – a freeform mix of calculations, text, drawings and equations in an electronic notebook. A number of these tools are available (including Maple, available via the APA website).

Modern calculation tools reproduce the design metaphor of hand calculations.

Additionally, these modern software tools can do math that is improbably difficult to do by hand (for example, FFTs, matrix computation and optimization) and connect to CAD packages.

For example, Brunel could have designed the chain links on the Clifton Suspension Bridge, and updated the dimensions of a CAD diagram, while still maintaining the readability of hand calculations, all from the same electronic notebook.

That seems like a smarter choice.

Would I go back to the physical notebooks that Brunel diligently filled with hand calculations? Given the scrawl that I call my handwriting, probably not.

## Possible Bug in Maple ?

by: Maple

Has anyone tried to run the following in Maple command-line mode (i.e. in terminal window, type "maple" to start it without the graphic interface),

"

expr1:=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16+t17+t18+t19+t20+t21+t22+t0-t0+t23;
expr2:=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16+t17+t18+t19+t20+t21+t22+t0-t0+t23;
print(expr1-expr2);

"

Surprisingly, I didn't get "0" with my Maple 17 (under Linux platform) or 18 (under Mac OSX platform). Can anyone help me confirm this?

## MaplePrimes Update - New Tools for Spam, Saving...

by: MaplePrimes

Earlier today, we published some changes to MaplePrimes in order to add some new features that have been requested by our users.

Delete as Spam

Users with a reputation score above 500 have always had the ability to remove inappropriate or spam posts using the Delete function. As of this update, a Delete as Spam feature has been added that provides some additional capabilities targeted directly at the spam messages that are posted on MaplePrimes. When this feature is used, the following occurs:

• The message is deleted
• The author's account is marked as a spam account and blocked
• All other messages from the author are removed

Since this is a powerful feature, there are also some safeguards in place to prevent accidents from occurring:

• Only users with a reputation less than 10 can be removed in this way
• In the event that content is ever inadvertently removed, MaplePrimes administrators have the ability to restore content removed in this way.

Hopefully, these features make it much more difficult for spam accounts to get traction on MaplePrimes, and also reduce the time involved in policing the spam that is posted.

Draft Messages

You now have the ability to save a message as a draft so that you can come back to it later. All draft messages are saved in your profile and can be managed by clicking the account option at the top of every page.

Some powerful searching capabilities have been added to MaplePrimes with the addition of a new Advanced Search feature. Using this search allows you to quickly find questions or posts within a particular date range, by author and/or by tag.

Save page as PDF

We know that some of our members maintain personal archives of MaplePrimes content, and to make this easier, we have added a feature that generates a PDF copy of a page. This option is available in the More area that appears under every post and question.

Your MaplePrimes user profile page has always included archives of your posts, questions and answers. With today's update, replies are also available.

In addition to these, some other minor changes & fixes were made.

As always, thank you very much for your dedication to MaplePrimes and for your suggestions. We hope that you find these features useful and look forward to other ideas that you have!

## They're watching...

by: MaplePrimes

Like most companies today, Maplesoft monitors its website traffic, including the traffic coming to MaplePrimes. This allows us to view statistical data such as how many total visits MaplePrimes gets, how many unique visitors it gets, what countries these visitors come from, how many questions are asked and answered, how many people read but never respond to posts, etc.

Recently one of our regular MaplePrimes users made the comment that MaplePrimes does not reach new Maple users. We found this comment interesting because our data and traffic numbers show a different trend. MaplePrimes gets unique visitors in the hundreds of thousands each year, and since its inception, it has welcomed unique visitors in the many millions. Based on these unique visitor numbers and the thousands of common searches specifically about Maple that people are doing, we can see that many of these unique visitors are in fact new Maple users looking for resources and support as they begin using Maple. Other visitors to MaplePrimes include people who use Google (or other search engines) to find an answer to a particular mathematics or engineering question, regardless of what mathematics software they choose to use, and Google points them to MaplePrimes. There are some popular posts that were written months, even years ago, that are still getting high visitor views today, showing the longevity of the information on MaplePrimes.

MaplePrimes gets the majority of its visible activity from a small number of extremely active members. In public user forums around the world, these types of members are given many names – power users, friendlies, evangelists. Every active public user forum has them. On MaplePrimes, it’s this small number of active members that are highly visible. But, what our traffic data reveals is the silent majority. These people, many of them repeat visitors, are quietly reviewing the questions and answers that our evangelists are posting. The silent majority of MaplePrimes visitors are the readers; they are the quiet consumers of information. For every person that writes, comments on, or likes a post, there are thousands more that read it.

Here are a few more MaplePrimes traffic data points for your reference:

• MaplePrimes is very international and draws people from all around the world. Here are the top 10 countries where the most MaplePrimes visitors come from:
1. USA
2. India
4. Germany
5. China
6. United Kingdom
7. Brazil
8. Australia
9. France
10. Denmark
• Here are the top 5 keywords people are using in their searches on MaplePrimes:
1. Data from plot
2. Physics
3. Sprintf
4. Size of plot
5. Fractal
• MaplePrimes is growing at a very fast rate: Traffic (visitors to the site) and membership size is growing at nearly double the pace it was last year. The total number of posts and questions this year is also much higher compared to the same timeframe last year.
• Our top 5 MaplePrimes members have each visited MaplePrimes more than 1200 times and viewed a combined total of more than 10,000 pages (that is total page views, not unique page views). Our top 25 MaplePrimes members have visited at least 250 times each (many of them nearly 1000 times each) and our top 50 MaplePrimes members have visited a combined total of over 23,000 times, visiting nearly 200,000 pages. Thank you! We’re glad you like it. :-)

## Delay differential equations - Maple is a leader

Maple's dsolve numeric can solve delay ODEs and DAEs as of Maple 18. However, if I am not wrong, it cannot solve delay equations with a time dependent history. In this post I show two examples.

Example 1:

y1(t) and y2(t) with time dependent history. Use of piecewise helps this problem to be solved efficiently. Hopefully Maple will add history soon in its capability.

Example 2:

This is a very a complicated stiff problem from immunology. As of now, I believe only Maple can solve this (other than RADAR5 from Prof. Hairer). Details and plots are posted in the attached code.

Let me know if any one has a delay problem that needs to be solved. I have tested many delay problems in Maple (they work fine). The attached examples required addtional tweaking, hence the post.

I want to take this opportunity to congratulate and thank Maple's dsolve numeric/delay solvers for their fantastic job. Maple is world leader not because of example1, but because of its ability to solve example 2.

 > restart;

This code is written by Dayaram Sonawane and Venkat R. Subramnian, University of Washington. You will need Maple 18 or later for this. For those who are wanting to solve these problems in earlier versions, I can help them by offering a procedure based approach (less efficient).

Example1 The first example solved is a state dependent delay problem (http://www.mathworks.com/help/matlab/math/state-dependent-delay-problem.html).

 > eq1:= diff(y1(t),t)=y2(t);
 (1)
 > eq2:=diff(y2(t),t)=-y2(exp(1-y2(t)))*y2(t)^2*exp(1-y2(t));
 (2)

Both y1(t) and y2(t) have time dependent history (y1(t)=log(t) and y2(t)=1/t, t<-0.1). If I am not mistaken one cannot solve this directly using Maple's dsolve numeric command. However, a simple trick can be used to redefine the equations for y1(t) and y2(t) as below

 > eq3:=diff(y1(t),t)=piecewise(t<=0.1,1/t,y2(t));
 (3)
 > eq4:=diff(y2(t),t)=piecewise(t<=0.1,-1/t^2,-y2(exp(1-y2(t)))*y2(t)^2*exp(1-y2(t)));
 (4)

The problem is solved from a small number close to t = 0 (1e-4) to make Maple's dsolve numeric remember the history till t = 0.1

 > epsilon:=1e-4;
 (5)
 > sol:=dsolve({eq3,eq4,y1(epsilon)=log(epsilon),y2(epsilon)=1/epsilon},type=numeric,delaymax=5):
 > with(plots):
 > odeplot(sol,[t,y1(t)],0.1..5,thickness=3,axes=boxed);
 > odeplot(sol,[t,y2(t)],0.1..5,thickness=3,axes=boxed);
 > sol(5.0);log(5.0);1/5.0;
 (6)

Tweaking the tolerances and epsilon will get the solution even more closer to the expected answers.

 >
 >

Example 2

The next problem discussed is very stiff, complicated and as of today, according Professor Hairer (one of the world's leading authority in numerical solutions of ODEs, DAEs), cannot be solved by any other code other than his RADAR (5th order implicit Runge Kutta modified for delay equations, Guglielmi N. and Hairer E. (2001) Implementing Radau IIa methods for stiff delay differential equations. Computing 67:1-12). This problem requires very stringent tolerances. For more information read, http://www.scholarpedia.org/article/Stiff_delay_equations. I can safely say that Maple can boast that it can solve this delay differential equation by using a switch function (instead of Heaviside/picecewise function). Code is attached below and results are compared with the output from RADAR code.  Note that dsolve/numeric is probably taking more time steps compared to RADAR, but the fact that Maple's dsolve numeric solved this model (which cannot be solved in Mathematica or MATLAB[needs confirmation for MATLAB]) should make Maple's code writers proud. It is very likely that we will be trying to submit an educational/research article on this topic/example soon to a journal. For some weird reasons, stiff=true gives slightly inaccurate results.

 > restart:
 >
 (7)
 (8)
 > eq[1]:=diff(y[1](t),t)=-r*y[1](t)*y[2](t)-s*y[1](t)*y[4](t);
 (9)
 > eq[2]:=diff(y[2](t),t)=-r*y[1](t)*y[2](t)+alpha*r*y[1](y[5](t))*y[2](y[5](t))*H1;#Heaviside(t-35);
 (10)
 > eq[3]:=diff(y[3](t),t)=r*y[1](t)*y[2](t);
 (11)
 > eq[4]:=diff(y[4](t),t)=-s*y[1](t)*y[4](t)-gamma1*y[4](t)+beta*r*y[1](y[6](t))*y[2](y[6](t))*H2;#Heaviside(t-197);
 (12)
 > eq[5]:=diff(y[5](t),t)=H1*(y[1](t)*y[2](t)+y[3](t))/(y[1](y[5](t))*y[2](y[5](t))+y[3](y[5](t)));#eq[7]:=y[7](t)=HH(t);
 (13)
 > eq[6]:=diff(y[6](t),t)=H2*(10.^(-12)*0+y[2](t)+y[3](t))/(10.^(-12)*0+y[2](y[6](t))+y[3](y[6](t)));
 (14)
 > H1:=1/2+1/2*tanh(100*(t-35));H2:=1/2+1/2*tanh(100*(t-197));
 (15)
 > alpha:=1.8;beta:=20.;gamma1:=0.002;r:=5.*10^4;s:=10.^5;
 (16)
 > seq(eq[i],i=1..6);
 (17)
 > ics:=y[1](0)=5.*10^(-6),y[2](0)=10.^(-15),y[3](0)=0,y[4](0)=0,y[5](0)=1e-40,y[6](0)=1e-20;
 (18)
 > #infolevel[all]:=10;
 > sol:=dsolve({seq(eq[i],i=1..6),ics},type=numeric,delaymax=300,initstep=1e-6,abserr=[1e-21,1e-21,1e-21,1e-21,1e-9,1e-9],[y[1](t),y[2](t),y[3](t),y[4](t),y[5](t),y[6](t)],relerr=1e-9,maxstep=10,optimize=false,compile=true,maxfun=0):

note that compile = true was used for efficiency

 > t11:=time():sol(300);time()-t11;
 (19)
 > with(plots):
 (20)
 (21)

Values at t = 300 match with expected results.

 > p[1]:=odeplot(sol,[t,log(y[1](t))/log(10)],0..300,axes=boxed,thickness=3):
 > display({pr[1],p[1]});
 > p[2]:=odeplot(sol,[t,log(y[2](t))/log(10)],0..300,axes=boxed,thickness=3,numpoints=1000):
 > display({pr[2],p[2]});
 >
 > p[3]:=odeplot(sol,[t,log(y[3](t))/log(10)],0..300,axes=boxed,thickness=3):
 > display({pr[3],p[3]});
 > p[4]:=odeplot(sol,[t,log(y[4](t))/log(10)],197..300,axes=boxed,thickness=3,view=[197..300,-9..-5]):
 > display({pr[4],p[4]});
 > p[5]:=odeplot(sol,[t,y[5](t)],0..300,axes=boxed,thickness=3):
 > display({pr[5],p[5]});
 > p[6]:=odeplot(sol,[t,y[6](t)],0..300,axes=boxed,thickness=3):
 > display({pr[6],p[6]});

I happen to just have a look at mathematica's imagedeconvolve function http://reference.wolfram.com/language/ref/ImageDeconvolve.html .  I had a look at the Examples and saw how a very blurred image of Neil Armstrong standing on the moon with the lunar lander was deconvolved into some really amazing detail.

I don't believe that image could deconvolve into what they show on that page, It's somewhat misleading.

The only way that deconvolved image could have such great detail is the blurred image used was most likely convolved from the detailed image.

## Comparisons

by: Maple

Hello all ,

Having read a recent post about comparing Maple and Mathematica I'd like to throw my 2 cents (FWIW).

It is *silly* (not to say stupid) to compare these two softwares.

Maple can do "things" that Mathematica can't. For examples Differential Geometry, Lie Algrebra, covariant derivative and the like.

And Mathematica Manipulate command is far better than Maple Explore (just another exemple).

I have being using Maple since Release V.2 (1992) and Mathematica since Release 1.1.a (1991).

I use both of them on a daily basis and I *LOVE* them both.

Inputs are welcome :-)

Kind regards to all,

Jean-Michel.

## Maple 2015.1

by: Maple 2015

I wanted to let everyone know that there is a Maple 2015 update available. Maple 2015.1 provides:

• Support for high-resolution monitors (e.g. 4K, UHD)
• Updated translations for Brazilian Portuguese, French, Japanese, and Simplified Chinese
• Enhancements to the Explore command
• Improvements to the DataSets package
• Updates to the Microsoft Excel plug-in
• Enhancements to unit handling
• A variety of improvements to the math engine, interface, and documentation

To get this update, you can use Tools>Check for Updates from within Maple, or visit Maple 2015.1 Downloads.

If you are a MapleSim 2015 user, you already have this update, as it was part of the MapleSim 2015 installation.

Kim

## Building a model of the Star Wars BB-8 Droid in...

by:

The trailers for the new Star Wars movie (Star Wars: The Force Awakens) introduced a new Droid called BB-8. This curious little guy features a spherical body and a controlled instrumented head. More recently, the BB-8 droid was showcased in a Star Wars celebration event and to many peoples' surprise it is real and not a CGI effect!

We have a Sphero robot from Orbotix here at the office, and there was an immediate connection between BB-8 and the Sphero. All that remains is to add the head!

Many have already put together their version of the BB-8, but I wanted to have a physical model that I can play with in a virtual environment and explore some design options.

Preparation:

To build a model of BB-8 like robotic system in MapleSim (Maplesoft's physical modeling software environment), I first needed a couple things in place before going forward:

1. A few simple CAD shapes (half-sphere, wheels)

2. A component to represent the contact between two spheres (both outside contact and inside contact)

I used Maple’s plottools package to build the CAD files I needed. First a half-spherical shape:

Then a wheel:

The next step was to create the contact component in MapleSim. I used a Modelica custom component to bring together vector calculations of normal and tangential forces with a variety of options for convenience into one component:

Build the model:

Then we add two wheels inside it, and a hanging mass to keep the reference axis vertical when the wheels turn:

Learning from published diagrams showing the internal mechanism of a Sphero, another set of free wheels improves the overall stability when motion commands are given to the two active wheels:

Now this model can be used to move around the surface by giving speed commands to the individual motors that drive to the two bottom wheels. What is needed next is the head and the mechanism to move it around.

Since the head can move almost freely, independent of body rotation, it has to be controlled via magnetic contacts and a controlled arm.

First, we add the control arm:

Now we need to build the head.

The head has an identical triangle to the one at the end of the control arm. At each vertex there is a ball bearing that would slide on the surface of the main spherical body without friction. The magnetic force between the corresponding vertices of the two triangles is modeled via the available point-to-point force element in MapleSim.

Once assembled, the MapleSim model diagram looks like this:

...and our BB-8 droid looks like this:

Seeing the BB-8 in action:

Now that we have constructed our droid in MapleSim, we can animate and see it in action!

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