1. Geometric reasoning about mechanical assembly By Randall H. Wilson and Jean-Claude Latombe Henrik Tidefelt

2. Topics Automatic generation of assembly algorithms Characterization of the complexity of assembly designs

3. Assembly tree An assembly algorithm (plan) is constructed by splitting the target into smaller and smaller subsystems. This yields a partial order in time of assembly instructions.

4. Assembly planning using the generate-and-test strategy Relax some constraints to come up with candidate algorithms (assembly trees). At least, the constraints imposed by the manipulating system are ignored. Search the candidates for globally feasible ones. (Motion planning including the manipulating system.)

5. NDBG The non-directional blocking graph represents how the parts in an assembly are constraining each other. It is useful for efficient generation of candidate algorithms during assembly planning, and also for complexity evaluation of mechanical assemblies. The NDBG - and hence its interpretation - is a function of the family of motions that is considered.

6. DBG The directional blocking graph only takes into consideration motions in a particular direction d. There is one node per part, and an arrow from part p 1 to part p 2 if p 2 is blocking p 1 in the direction d. DBGs for infinitesimal translation along d

7. DGB A strongly connected component can not be (dis)assembled along the direction d.

8. DBG A subset with no outgoing arcs is locally free to translate in the direction d, But there is no guarantee that this cut corresponds to a globally feasible assembly plan.

9. NDBG S, the set of all directions, can be divided into intervals over which the DBG is constant. These intervals are called regular regions. The NDBG is a structure associating each regular region with a corresponding DBG.

10. Computing the NDBG Given an assembly of parts, we can find the regular regions by cutting S in every direction that is parallel with an edge in contact with another part. Every cutting direction is a regular region, as are the open intervals separated by the cuts. Compute DBG for each regular region.

11. Computing the DBG The DBG is represented as an n by n adjacency matrix, where n is the number of parts. First, clear the whole matrix. Then, pick an arbitrary direction d in the region and evaluate each edge contact.

13. Time complexity Let c be the number of edge contacts. There are O(c) regular regions. Each DBG is computed by considering c edge contacs. Computing a NDBG in this way takes O(c 2 ) time.

14. Improvements Instead of considering all c directions of edge contact, we can look at each pair of parts in contact and find the direction along which they can slide along each other. These are the only directions that need to be considered. Let there be r pairs of parts in contact. Finding the r directions takes O( c + r log r ) time, and after addition of the r 2 time it takes to compute all adjacency matrices, the total time becomes O( c + r 2 ). It is guaranteed that r ≤ c, and in many cases r << c.

15. More improvements It is possible to exploit the fact that adjacent DBGs are similar, resulting in an O( r 2 ) algorithm. If the application only makes use of one DBG at a time, and can do that in an order so that subsequent regular regions are also adjacent, only one DBG needs to be stored at any time.

16. Other kinds of NDBGs So far, we have only seen NDBGs representing local freedom of translation in the plane, i e limitations on infinitesimal translations in the plane, but the NDBG is suitable for other kinds of motions too: Infinitesimal generalized motions (local freedom of translation and rotation) Infinite translations Extension to 3D

17. Complexity evaluation Aiming at supporting the designer of mechanical assemblies to create products that are easy to mass-produce and maintain. Compare with the importance of knowing the time and space complexity of a computer algorithm. To automate the complexity evaluation we need algebraic complexity measures.

18. Complexity measures p-handed Monotonic? m-prismatic Stack? Length Linearizable? Degree of form closure There exist an algorithm where each instruction involves at most p + 1 moving subsets, and p - 1 hands are not sufficient.

19. Complexity measures p-handed Monotonic? m-prismatic Stack? Length Linearizable? Degree of form closure Every instruction moves a subassembly to its final position relative some other subassembly.

20. Complexity measures p-handed Monotonic? m-prismatic Stack? Length Linearizable? Degree of form closure There exist an algorithm where the instructions move each subset in a way that can be described by a sequence of at most m extended translations. m - 1 is not enough.

21. Complexity measures p-handed Monotonic? m-prismatic Stack? Length Linearizable? Degree of form closure Length of longest sequence of instructions. 1-handed, moving only one part per instruction. Fingers needed to grasp subassemblies with form closure.

22. Example The assembly has three parts, so if it is admissible, it will be at least 2-handed monotonic. Given that we may only do translations, is it 1- handed monotonic? Given translation and rotation?

23. 2-handed: monotonic

24. 1-handed translations: not monotonic

25. 1-handed translation and rotation: monotonic

26. NDBGs and complexity evaluation All 1-handed assembly algorithms that are correct for infinitesimal translations can be extracted from the assembly’s NDBG. This leaves a polynomial set of algorithms to try to see if the assembly is 1-handed monotonic prismatic, or linearizable for translations.

27. NDBGs and complexity evaluation The NDBG can be used to compute an upper bound on the number of fingers required to give a subassembly form closure: By identifying loose parts of the assembly in the NDBG, we can find appropriate placed to place fingers on. We add a finger at a time until the assembly has the form closure property, updating the NDBG after each modification. (Force closure might be more practical.)

28. NDBGs and complexity evaluation All 1-handed monotonic 1-prismatic assembly algorithms can be extracted from the NDBG of infinite translations. The product is a stack iff such an algorithm can be extracted from a single DBG.

29. Conclusions The NDBG can be used both for generation of assembly algorithms and for complexity evaluation of mechanical assemblies. Assembly planning using NDBGs is done in generate-and-test fashion. Complexity evaluation can help designers design products that are suitable for mass- production and easy to maintain.