[HN Gopher] Robotics 501: Mathematics for Robotics
___________________________________________________________________
Robotics 501: Mathematics for Robotics
Author : rg111
Score : 215 points
Date : 2022-01-27 11:50 UTC (11 hours ago)
(HTM) web link (github.com)
(TXT) w3m dump (github.com)
| overkalix wrote:
| Seems a pretty standard linear algebra course, nothing that
| justifies the title...
| funnym0nk3y wrote:
| It seems very shallow too.
| pfortuny wrote:
| Honest question (am a prof. of Maths for engineering at a Spanish
| Univ. and am wondering about this a lot):
|
| What is the point of explaining, say the QR factorization or
| iterative methods like Gauss-Jordan to _engineers_? I think of
| them as the method of integration of rational functions (the
| complete method, not the basic cases): totally useless _today_...
|
| I understand, for example, the Jordan decomposition theorem in a
| Linear Algebra course _because_ it gives a clear-cut description
| of a linear map and the meaning of eigenvalues /eigenspaces, etc.
|
| Anyone cares to share? Thanks in advance.
| rck wrote:
| In robotics specifically, understanding how methods like QR
| factorization work (and fail) is necessary to debug robot code,
| even if roboticists rarely have to implement QR factorization
| from scratch.
|
| Take a look at GTSAM, a library used widely in robotics for
| mapping and localization. To read the intro tutorial for the
| library, you need to understand most of the topics listed in
| the OP repo:
|
| https://gtsam.org/tutorials/intro.html
|
| If anything, the Michigan course doesn't cover enough, since
| it's missing Lie theory, which is becoming standard in robotics
| code these days.
| slingnow wrote:
| The answer is simple: this course is taught at a university, so
| they have to fill the time with _something_. And if that
| something is a rote method that lends itself well to testing,
| even better! It matters little if actual engineers in the field
| use it (I've done robotics for years, and knowing that Gauss-
| Jordan exists has helped me precisely zero times).
| pfortuny wrote:
| Come on, do not simplify like this (I am teaching it and
| precisely because I am worried, I try to fill the course with
| _relevant_ topics).
| pfortuny wrote:
| *Replying to the answers received as of 17:02 20220127 GMT+1.*
|
| Thanks A HUGE LOT to all and sundry for taking the time.
|
| I am the first one to understand that theory is the only way to
| really grasp a problem (this is why differential equations and
| calculus and linear algebra are key in the education of an
| Engineer). This is something I guess all of you agree on (as a
| matter of fact, I am a pure mathematician by education).
|
| My problem with *specific algorithms* is that they are deemed
| to be obsolete, unless they _are inherently theoretical_. In
| this sense:
|
| a) Gauss reduction method (some of you have mistaken this for
| the iterative Gauss-Jordan method, I guess): this is not just
| _a method_ but a way to understand a linear map (and a system
| of equations) in a much simpler way, and of getting information
| by itself (i.e. the eigenvalues). Same for the Jordan
| decomposition result (but this is a bit overkill to me).
|
| b) Newton-Raphson: the most important use of the "derivative as
| approximation" that exists, apart from showing the importance
| of iterations and stability.
|
| c) The condition number, and (in)stability of linear systems:
| this is a pure theoretical notion (nobody computes the
| condition number because if you can, then you can also invert
| the matrix) which has very important *applications as
| understanding*.
|
| However:
|
| a) Gauss-Jordan (i.e. the iterative method): I understand that
| it may be good to teach it as a tool to understand convergence
| and iterations but (you won't believe this): here they ask the
| students to *perform* iterations of this method BY HAND WITH A
| CALCULATOR...
|
| b) QR: I never understood teaching this to undergrad engineers
| (it is just the Gauss reduction method with a little care when
| your matrix is symmetric). What interest does this have _in
| general_ for an engineer?
|
| Etc...
|
| Compare the two above with, for instance, the method of
| integration of rational functions in its most general form,
| which was taught in Spain in schools of engineering until
| twenty years ago. Nobody sees this as relevant any more, as it
| adds _absolutely nothing_ to the understanding of the integral,
| and is just a waste of time (Wolfram|Alpha does it for free)...
|
| By the way, and you will not believe this either (I have been
| teaching this course several years by now). Where I work it is
| *in the first year*. This hurts a lot, as my students have
| absolutely no way to apply any of this to any engineering
| problem.
| pfortuny wrote:
| OH MY!
|
| I intended to say Gauss-Seidel and said Gauss-Jordan!
|
| What a mess I have created. Sorry to all for the confusion.
|
| Of course, Gauss' reduction method is key to understanding
| systems/linear maps. Gauss/Seidel is totally different,
| though.
|
| My apologies.
| Tainnor wrote:
| Even as a pure maths student, I've rarely found it useful to
| have to carry out complicated computations in exam settings.
| Integrations methods I can even understand up to a point as
| it is useful for pattern matching - but I had to actually
| carry out the simplex method on paper in one exam, and worse,
| I had to memorise several variants of it with different
| pivoting strategies. Of course, I barely remember how that
| algorithm works by now.
|
| I guess one reason why such computations are asked for in
| exams, though, is because it gives people something concrete
| to study so that if they prepare well, they can score some
| points. If you only have "applied knowledge" exams, they get
| harder. And somehow, society wants to perpetuate the notion
| that you can achieve anything if you put in enough effort (no
| matter whether that effort is actually valuable).
|
| In an ideal world, we could instead maybe give students a
| high level description of the algorithm, have them carry it
| out explicitly on _extremely_ simple input just to see it
| once, and then maybe ask them to implement the algorithm in a
| computer program. Sadly, at least for pure maths students,
| the coding skills are typically low even when there are
| explicit CS requirements, don 't know what it's like for
| engineering students.
| Tainnor wrote:
| I think the Gauss-Jordan method has some reason to be taught:
|
| - Most people have probably solved systems of equations by hand
| in school. G-J basically just systematises this process by
| abstracting away the unnecessary details and it has a clear
| sequence of steps (although you can still choose different
| pivots). I think it would not be satisfying to never see that.
|
| - G-J can be done by hand on simple systems.
|
| - It is a rather simple algorithm, but it still has some edge
| cases to consider (what if you have a 0 in pivot position?).
| I've seen a number of people try to implement G-J (we used it
| as a coding challenge and, yes, this was actually job-related
| in this case ;)) and fail on particular examples, e.g. stopping
| iteration too early after skipping a column.
|
| - The algorithm runs in polynomial time (assuming floating
| point arithmetic; it's unfortunately not polynomial for
| arbitrary precision integers), so it's efficient. Knowing that
| a system of linear equations can efficiently be solved is, I
| think, rather important.
|
| - you can discuss numerical stability in the context of
| different pivot strategies
|
| - other algorithms use it as a building block or are basically
| a variation of it (LR factorisation, matrix inversion, simplex
| algorithm, etc.)
|
| Of course, you could probably just tell people that instead of
| teaching them the algorithm, but I would think that it sticks
| better if you've actually seen it and it's not _that_ hard of
| an algorithm (and I don 't think you need to require people to
| memorise it either).
|
| If you'd ask about teaching Jordan decomposition to engineers I
| would agree. :)
|
| (But I've also never taught anyone, especially not engineers.
| So this is just what makes sense to me personally.)
| slingnow wrote:
| Pay attention to all of the answers you're receiving. It's like
| they're coming from the University itself! Not one single
| concrete example of where someone has used GJ in the field,
| just a bunch of the same excuses and hypotheticals regurgitated
| in different ways.
| gmadsen wrote:
| Just want to point out, this is a graduate course(although 1st
| year). Many students taking it are phd students, who at
| Michigan can do quite theoretical research. Its not uncommon to
| take graduate math and cs courses as an engineering phd at
| Michigan
| pfortuny wrote:
| Yes, I understood that from the 5 at the beginning. OK, so it
| is not a Masters but something directed to PhD students then?
| gmadsen wrote:
| I wouldn't say that, it is a mix of masters/phd. the
| robotics department is quite new, the overall curriculum is
| pretty rigorous mathematically, but also a lot of lab work.
|
| this is more of a catch all to make sure everyone is able
| to handle the rest of the program, since depending on
| undergrad school and country, mathematical maturity seems
| to be a large spectrum for incoming robotics students
| pfortuny wrote:
| OK, thanks!
| BeetleB wrote:
| Although I agree with your general point, but...
|
| Just recently I solved a problem in two different ways (i.e.
| two methods that led to an equation of the form Ax=b), and
| after I'd done that, I wondered if they were merely equivalent.
| The first thing that popped into my head was to reduce to Row
| Reduced Echelon Form.
|
| There are probably other ways to show equivalence, but here is
| _one_ engineer who did use it.
|
| (Albeit for a hobby, not a job).
|
| Sadly, most engineers (including me) do not use on the job over
| 90% of the math we learn in university. And when I actively
| did, there were disincentives to do so (e.g. no rewards for
| using them, culture just doesn't value them, etc).
| pfortuny wrote:
| Yes, but that method is much more than "a method", as I
| explain in my other reply.
|
| Thanks though!
| neel_k wrote:
| It sort of depends upon the course. This particular syllabus
| looks like a really standard linear algebra course (I have no
| idea what is robotics-specific about it), and so of course
| Gauss-Jordan would show up there.
|
| More generally, though, I think there are two main reasons to
| put algorithms like Gauss-Jordan in the engineering curriculum.
|
| 1. While it would be nice to treat linear algebra solvers as a
| perfect black box, in practice this does not work. Engineers
| have to be aware of numerical stability issues, how to diagnose
| when this is an issue, and how to reformulate their routines in
| a way that resolves the problem. And to do this, they need to
| know the library of techniques.
|
| A reasonable way of teaching this is to teach Gauss-Jordan,
| then showing how it goes badly awry, and then showing how
| things like pivoting can fix it.
|
| 2. Personally, though, I find the topic of numerical stability
| to be a little bit depressing, since it focuses on all the ways
| computers don't work!
|
| To take a more positive view, a huge fraction of the algorithms
| in an undergraduate CS course -- from finite automata to
| parsing to relational algebra to graph traversals -- can be
| understood as basically doing linear algebra using modules over
| different semirings (rather than just the reals). Eg, for all-
| pairs shortest paths, the Floyd-Warshall algorithm is doing an
| LU decomposition, and Kleene's algorithm is doing Gauss-Jordan.
|
| Not every student will enjoy this, but for the ones who are
| algebraically minded, it's really exciting to be able to offer
| them a unified perspective. And then you can show them the
| GraphBLAS library!
| a_zaydak wrote:
| I really agree with your comments; especially number 1. Often
| I can't use some black box implementation of an solver (or
| other algorithm for that matter) without some modifications.
| Numerical stability is a big one but also just performance.
| Sometimes the mathematically correct way of doing something
| is not always the best in practice. Short cuts and
| approximations can provide huge benefits. It is difficult to
| make those modifications without understanding the inner
| workings of the original method.
| Tainnor wrote:
| One example is that GJ stops being generally efficient in
| arbitrary precision settings. Many people will never have
| to deal with this, but if you are doing cryptography, it
| matters.
|
| For example, while you can use GJ to calculate the
| determinant of a matrix, this can easily become
| exponential, and for integer matrices (or generally for
| matrices over division rings), there is an alternative
| method (the Bareiss algorithm) that is actually efficient.
| Tainnor wrote:
| > 2. Personally, though, I find the topic of numerical
| stability to be a little bit depressing, since it focuses on
| all the ways computers don't work!
|
| Maybe a way to more positively reformulate this would be:
| There is no a priori reason to assume that floating point
| numbers are well behaved. The fact that we were able to come
| up with a structure so that it approximates real numbers
| adequately, that arithmetic operations on it are fast (which
| they aren't for infinite precision) and that, if we design
| the algorithms correctly, errors are well-behaved, is an
| astonishing feat of engineering.
| idealmedtech wrote:
| Deep understanding of the mathematics behind a given problem
| space can help you later in the applications. So maybe they'll
| never have to actually implement Gauss Jordan (and they
| shouldn't), but the core ideas of normalization and reduction
| are very important to any sort of numerical methods and come up
| again and again.
|
| As a somewhat related example, our engineering college required
| a calculus based probability course that probably 95% of the
| engineers dreaded. Conditional probability, multivariable PDFs
| etc. A few years after college, I had the (dis)pleasure of
| needing to generate random values from an unknown PDF, given
| only the cumulative distribution. To a statistician, the
| obvious answer is to use the inverse CDF (a simple interpolated
| lookup table based on the known CDF) and a uniform random, but
| an engineer who didn't fully grok the probability basics would
| probably have just hacked around it, trying different
| approximations until something stuck.
|
| Neither approach is right or wrong, but sometimes simply
| understanding the essential material and methods gives you
| unforeseen insights later on.
| BeetleB wrote:
| > Deep understanding of the mathematics behind a given
| problem space can help you later in the applications.
|
| The commenter isn't against that - he makes this clear in
| other comments.
|
| He's arguing that Gauss Jordan _doesn 't_ give deep
| understanding.
| enriquto wrote:
| > What is the point of explaining, say the QR factorization
| (...) engineers?
|
| As a personal data point, I'm a mathematician that regularly
| teaches the QR and SVD factorizations to engineers. Not because
| I find them interesting, but because they need it for their
| daily jobs and they ask me to.
|
| More precisely, they are engineers working in the geometrical
| modelling of satellite sensors, doing "computer vision" from
| satellite images. The QR and SVD factorizations of 3 and
| 4-dimensional matrices are essential tools for the bundle
| adjustment steps, to represent the position and orientation of
| each satellite and the transformations between them. A crucial
| idea (for them) is that A=QR is a unique decomposition of A
| which is also continuous (the coefficients of Q and R depend
| smoothly on the coefficients of A). On the contrary, the SVD
| decomposition is neither unique nor continuous. Since the SVD
| provides an easier-to-use decomposition, you want to use it
| whenever possible, but being careful to never assume continuity
| nor uniqueness; if you need these properties then you pick the
| QR.
| pfortuny wrote:
| Right, in this case you are totally right.
|
| In my school however, it is explained for general-size
| systems. With this I do have issues.
|
| Thanks for your feedback.
| PennRobotics wrote:
| The most mathematically-relevant stuff I remember was inverse
| kinematics and using numerical approximation to really cut down
| on computation when switching between coordinate spaces. This
| gets more important as the degrees of freedom increase.
|
| For instance, you'd like to know that when a state-space matrix
| is not full rank, you lose some control. In practical terms, a
| robot arm can usually move in any arbitrary Cartesian
| direction. A fully outstretched robot arm cannot.
|
| For computing inverse kinematics... It's much simpler for a
| processor to have a slow update cycle that computes how every
| motor's changes will affect the position of the end effector
| (robot "tip") and then take a numerical inverse and use that to
| figure out a time-local, space-local approximation of getting
| to your eventual goal. It's a helluva lot easier than solving
| exactly.
|
| I also recall some of the mathematics being relevant to
| collision detection (both in planning and execution stages),
| but the details are hazy. Basically, computation is expensive
| and convex hulls and precomputation can save a lot of cycles
| until it matters.
|
| Yet other linear algebra lands in the realm of computer vision,
| optimization, and finite-element analysis.
| pfortuny wrote:
| Yes, of course, but those are not _specific algorithms_
| (except possibly the Gauss reduction method which deserves a
| special treatment because it is not "just a method of
| computing" but "a way of understanding" by itself).
| PennRobotics wrote:
| Yep. I haven't really had much need for the techniques I
| learned in my graduate linear algebra course and found the
| in-class examples to really lack specific usefulness in the
| real world.
|
| I just looked up my old class: linear algebra basics;
| Gaussian/LU/Choleski decomp; determinants; normed spaces;
| condition number; iterative methods; Euclidean spaces; QR
| decomp; Hermitian geometry; Eigenscheissen; spectral
| theorems; finite elements method; SVD and pseudoinverses;
| quadratic optimization.
|
| Can confirm, totally useless in the non-research world. The
| most applicable task was computing spline curves. Even the
| Google Images result for "hermitian geometry" is mostly
| images of research papers. How is that real-world relevant?
|
| The useful robotics stuff for me is either already written
| as a library I can call or pseudocode I can find in AIMA.
| Then again, all of that stuff had to come from somewhere
| and receive the optimization treatment.
|
| I'd liken that entire Michigan mathematics course to the
| first week of my FEA course. "Here's how to calculate, by
| hand, a basic example of stress and strain in a very simple
| geometry using matrix operations. Cool, now that you see
| how much of a real hassle that is, never do it again
| because we have Ansys and Solidworks."
|
| -----
|
| On the other hand, a paper such as
| http://ras.papercept.net/images/temp/IROS/files/3131.pdf
| would seem totally inaccessible without a class like this
| Robotics 501. Maybe that constains good examples of the
| math being instructed. (Disclaimer: I didn't do more than
| glance at the course material and watch a few moments from
| the lectures.)
| Tainnor wrote:
| Can you explain the difference between GJ as "the Gauss
| reduction method"? Because I haven't seen these terms used
| before in a way that makes a difference between them.
| antman wrote:
| One needs to know the math to be able to understand how to
| transform the physical world to math so as to solve the
| problems. Math is not a computer module that can be used as a
| black box except in trivial cases.
|
| One needs to know the math to be able to understand how to
| transform the physical world to math so as to solve the
| problems.
|
| Example: Travelling salesman used to repair document gone
| through the document shredder. There isn't a book you can find
| this, either you understand the math principles or not.
| https://news.ycombinator.com/item?id=27713441
|
| Do you need to know the proofs though? Most of the time no, but
| sometimes you need to know more.
|
| Many proofs for example invert tables and in real code you
| better avoid it. Buy you need to know the Gauss Jordan to avoid
| the wtf when you crash into its common numerical instabilities.
| bunje wrote:
| QR factorization is cool because it is part of QR iteration and
| you can use it to find all zeros of a polynomial at once.
| pgorczak wrote:
| To expand on some of the other replies: maybe not QR and Gauss-
| Jordan directly, but understanding linear maps and subspaces is
| relevant in kinematics, especially kinematic chains like robot
| arms / manipulators.
|
| You can formulate a task in terms of some nonlinear function
| mapping the robot's joint positions to a desired "task space"
| and set a reference point or trajectory in that space (e.g.
| keep a tool perpendicular to a surface while following a line
| on that surface).
|
| Approaches for solving these tasks usually look at the Jacobian
| (inverse) of the task-function and use iterative methods to
| find a trajectory in joint-space that solves the task, either
| making a plan offline or as part of the control algorithm in
| real time.
|
| The singular value decomposition of that Jacobian has a couple
| of applications like
|
| - find the pseudoinverse for the solution
|
| - use the condition number to detect (proximity to)
| singularities
|
| - find a Tikhonov-regularized solution to avoid singularities
| (aka. "singularity robust inverse" or "damped least squares")
|
| - find a projection into the null space of the Jacobian or that
| of a lower-rank approximation and solve lower priority tasks in
| that space
|
| - use the range of the Jacobian to measure "manipulability" or
| agility in task space, e.g. as a secondary optimization
| objective
| [deleted]
| upbeat_general wrote:
| Having learned both the implementation details of various QR
| methods and seen QR pop up in proofs elsewhere I think that it
| can be helpful as a building block rather than useful
| standalone.
|
| That being said I've seen lots of algorithms that are only
| practical given that the matrix is in a special form for
| decomposition/inversion/etc and if you're trying to implement
| an existing algorithm, tweak it, etc. that knowledge is
| critical.
| cricci16 wrote:
| Well, if I understand your point you are refering to the fact
| that in many high level programming languages, the language
| itself is able to dected the most efficient method to do solve
| some linaer algebra problems, say solving linear systems.
| However QR factorization (or other factorizations) may be
| usefull when you have to solve different problems but with the
| same matrix, so that factorizing once and for all gives you an
| advantage (for example many undetermined linear problems [1]).
| In this case knowing something about factorizations, or what to
| use when, may be helpful
|
| [1]
| https://en.wikipedia.org/wiki/QR_decomposition#Using_for_sol...
| funnym0nk3y wrote:
| Well, I'm currently in my masters in electrical engineering in
| Germany and we are taught the same stuff. We even have a
| mandatory course about basic numerical algorithms.
|
| What exactly I can do with that I don't know
| mahdi7d1 wrote:
| I'm not even in electrical engineering but in Industrial
| engineering and I still had to take numerical algorithms
| course this semester. Also had to do the iterations by hand
| for the exam.
| pfortuny wrote:
| Yes, that is exactly my issue.
|
| It is unlikely you are going to _use_ the Gauss-Jordan method
| (if you find a linear problem of huge size, you are just
| going to plug it into Matlab or whatever) and it does really
| not give you any information on the problem.
|
| On the other hand, eigenvectors (and the principal components
| theorem) _do_ give you an idea.
|
| I reckon (as of now) that Theory is much more important than
| _methods_ : these are going to become obsolete very fast,
| whereas the geometric/qualitative/quantitative insights of
| theoretical results are what give you _understanding_.
|
| Thanks for the feedback.
| Archit3ch wrote:
| > if you find a linear problem of huge size, you are just
| going to plug it into Matlab or whatever
|
| It depends on your viewpoint. Are you going to be the user
| who calls the MATLAB function, or the engineer who
| implements a specific iterative method, optimizes it for
| new hardware, and so on?
|
| I am trying to make the jump into the second category.
| BeetleB wrote:
| > Are you going to be the user who calls the MATLAB
| function, or the engineer who implements a specific
| iterative method, optimizes it for new hardware, and so
| on?
|
| 99% of engineers will be in the former.
|
| Of the remaining 1% who would like to be the ones
| implementing this stuff, about 90% will not get the job
| that lets them do it.
|
| Of those who do get that job, likely over 50% of them
| will hate it.
|
| Speaking as one who's been through all of this.
| pfortuny wrote:
| No: the person who implements the method is _certainly_
| working alongside (or has handy) a say, mathematician
| /physicist/specialist in numeric computation. I would not
| ask "just an Engineer" (no offense) to understand how to
| properly implement such type of methods (mostly because
| they are full of special cases which require a
| specialist).
|
| Hope you make the jump to that job, which is surely
| interesting (and pretty hard).
|
| Thanks.
| baking wrote:
| One minute in to the first lecture: "We do theorems and proofs
| every single day, even today, and we never see anything practical
| on a robot."
| mishftw wrote:
| Prof Grizzle is very much about practical learning and has a
| pretty good sense of humor too. When I did undergrad at
| Michigan, I got the sense he was one of those exceptional profs
| who knows what they're doing + truly cares about the non-
| academics.
| addcninblue wrote:
| Genuinely curious: How does real analysis apply to robotics?
| pfortuny wrote:
| The very notions of continuity and extrema allow you to, for
| instance, tell if a mechanism is in general position or in a
| special case.
| erwincoumans wrote:
| See also the Modern Robotics course videos:
|
| https://youtube.com/playlist?list=PLggLP4f-rq02vX0OQQ5vrCxbJ...
|
| and free book:
| http://hades.mech.northwestern.edu/index.php/Modern_Robotics
|
| It also has a github repo:
| https://github.com/NxRLab/ModernRobotics
| hcrisp wrote:
| Modern Robotics adopts the Product of Exponentials (PoE)
| mathematical approach (also known as screw-theoretic) over
| Denavit-Hartenberg (D-H) for forward kinematics . Do robotics
| companies use PoE in their written software or is D-H still the
| practical choice?
| gmadsen wrote:
| Id say by leaps and bounds DH
|
| software in companies needs to be as easy to explain as
| possible to the widest audience.
|
| very few people have a firm understanding of exponential maps
| biotinker wrote:
| I work for Viam[0] and while all of our actual calculations
| are made using Spatial Vector Algebra[1], inputs can be made
| in a number of different ways and supporting DH is crucial
| for allowing pre-existing configurations to be easily brought
| over to our platform. SVA can also conveniently be easily
| used for both kinematics and dynamics.
|
| Personally I find SVA way more intuitive than DH or PoE.
|
| [0] https://www.viam.com/
|
| [1] https://link.springer.com/chapter/10.1007/978-1-4899-7560
| -7_...
| tonyarkles wrote:
| I did this course through Coursera a while ago and it's really
| fantastic!
| amelius wrote:
| This may be a fine course, but there seems to be nothing specific
| to robotics here. Not even control theory.
| quanto wrote:
| This is my disappointment as well. Mathematics for robotics is
| quite challenging due to control theory and non-linear
| optimization, which do not seem to be the focus of this course.
| ska wrote:
| I suspect this is a prerequisite course for the ones that are
| more topical.
|
| I'm not familiar with this particular department approach,
| but scanning the syllabus it seems like a pretty standard
| introductory "leveling" course you use in a graduate program
| taking in students from lots of different undergrad degrees.
| ianai wrote:
| " local vs global convergence in nonlinear optimization"?
| wcunning wrote:
| This is a class in the Robotics program at University of
| Michigan, which has a long and deep history in control theory
| and related topics. The detail you're missing is that the
| control theory classes are for both Robotics students and
| general EE's, ME's, Aerospace Engineers, etc, so they're not
| going to get combined with this. The thing this lets people
| avoid is the EE linear systems theory class, which broadly
| covers the same topics, probably along with the EE intro to
| probability class, since that's more for full on signal
| processing or stochastic control. It fits in nicely with the
| other curriculum components, just not as a general purpose math
| for robotics for a non-program student.
| Ostrogodsky wrote:
___________________________________________________________________
(page generated 2022-01-27 23:01 UTC)