# modeling mechanical systems with Lie algebras

I'm not a mathematician, so please forgive my obvious naivety.

I'm interested in generating a representation in vector space of a two dimensional armature, or set of rigid linked elements, with each element articulated by rotation about a point defined on the previous (acyclic) element, about an axis orthogonal to the plane of the armature. What I require is a basis vector and a set of transform matrices that cover the configuration space of the armature.

My understanding is that Lie algebras manipulate the identity homeomorphisms of Lie groups, in that, for example, the Lie algebra that generates the Lie group SO3 would define the local patches that contain the identity of a particular 3D rotation.

Also that an armature as described, with multiple articulations can be described in terms of a Lie algebra and that the algebra can be used to generate a representation and thereby a basis vector and transform matrices that cover the configuration space of the armature.

All my requirements are for mechanical systems, so constrained to special orthogonal groups.

If anyone has suggestions regarding methodology or sources to assist in this pursuit, I would be very grateful.

• (Be careful with the terminology "the Lie algebra that generates the Lie group $\mathrm{SO}_3$"; if a Lie algebra can be said to generate a Lie group in any sense, then the corresponding Lie group should be simply connected—in this case, a spin group.) – LSpice Jun 28 '17 at 11:10
• Hi LSpice, thanks for the resonse!.as I said I'm interested in continuous groups that apply to affine transforms of physical (newtonian, maco) systems, so simply connected. – J0N0 Jun 28 '17 at 11:27
• I appreciate the caveat, and that is really my key question. How do I whittle away the generality that is the goal of the algebraists and zero in on a tractable version with a specific application. – J0N0 Jun 28 '17 at 11:34
• But here your requirements are contradictory, because the special orthogonal groups are not simply connected. – LSpice Jun 28 '17 at 11:45
• yeah, i see what you mean. ie, a sphere (SO3) is simply connected but a torus (SO2xSO2) isn't! but i'm sure I've read that systems of mechanical actuators can be modeled by Lie algebras and that these algebras can be used to generate representations. (note that I'm more interested in applications than abstractions :) actually, what is the significance of the multiplication? – J0N0 Jun 28 '17 at 11:56

Probably your best bet is to read through Chapter 2 of A Mathematical Introduction to Robotic Manipulation by Murray et al. (free download at link). You seem to referring to "exponential coordinates" on SO(3) (see for example equation (2.16) on P.30). Although if your armature and rotations are confined to a plane, then you really want to consider direct products of the 2d Euclidean group $\mathrm{SE}(2) = \mathrm{SO}(2)\ltimes \mathbb{R}^2$ to cover your configuration space (one copy for each link). The 3D version $\mathrm{SE}(3)$ is covered in Appendix A.

Another place to look is Chapter 6 of Harmonic Analysis for Engineers and Applied Scientists by Chirikjian and Kyatkin.

• looks very useful, cheers! – J0N0 Jun 28 '17 at 19:46
• the harmonic analysis book looks helpful too. ...i'm hoping to develop a measure of information content of spatial motion of an armature, along similar lines to the compression of spatial signals used in the jpeg image compression algorithm (which encodes the hadamard transforms of signal windows). – J0N0 Jun 28 '17 at 19:55
• picked up a used copy of Chirikjian and Kyatkin online for ten bucks, so looking forward to its arrival! – J0N0 Jun 29 '17 at 12:34
• Murray et al. is exactly the material i'm looking for. the obvious plan for me is to work thru appendix A, which covers everything i need to know before i can really address the applied concepts. thanks ery much for this reference, much appreciated! – J0N0 Jun 29 '17 at 20:49
• Great, hope it's useful for you. BTW what I called "exponential coordinates" above are called "canonical coordinates of the first kind" in the Appendix of Murray et al. – user17945 Jun 29 '17 at 22:10

Chapter 2 of Edward Nelson's book on Tensor Analysis (https://web.math.princeton.edu/~nelson/books.html) contains a worked example of the relationship between a Lie algebra and a two dimensional car parking problem. It may help you to develop your analysis and requirements.

• this looks really interesting thanks! it's just the kind of material i'm after. i can work through an example like this with some confidence that it's applicable to the domain. i'll also be interested to see if this example can be related to nonholonomic kinematics. – J0N0 Jun 29 '17 at 12:32
• The book I suggested by Murray has a similar example - the car with N trailers, on P. 349. In fact, this example has almost the same configuration space as the armature you described (except the armature has one point fixed), so you might want to work through that example carefully. – user17945 Jun 29 '17 at 16:45
• excellent. i haven't had much time to look thru the book yet it but i did see many diagrams of physical mechanisms, so also exactly what i'm looking for. the nonholonomic interpretation isn't a priority. – J0N0 Jun 29 '17 at 20:24
• the system i'm looking at is arthropod locomotion and gait. the local kinematics of articulation of the skeleton around a foot in contact with the ground is holonomic, while the global kinematics of the movement of the whole animal can be modeled more like a skate or wheel. currently i'm more interested in the local kinematics, altho i am hoping to eventually link this to a global model. i'm interested in the control mechanisms that integrate these two aspects. – J0N0 Jun 29 '17 at 20:24