Geometric Reasoning About Mechanical Assembly By Randall H. Wilson and Jean-Claude Latombe Presented by Salik Syed and Denise Jones

Goals and uses of Assembly Planning Understand how objects are assembled, and their complexity. Can be used to guide designers and engineers to create more efficient products by allowing them to understand service and manufacturing costs at design time.

Generalized Motion Planning approach We can look at this as a general Motion planning problem. Problems: ● Complicated planning ● Key issues such as grasping are not known ● We don't know what the manipulator will be. ● We can instead use a “Virtual Manipulation system”

Hierarchical approach Instead we can abstract the notion of manipulators into a “virtual manipulation system”. Solve a succession of simplified but increasingly more realistic planning problems. Different levels of abstraction Free flying objects then at lower levels abstractions of different grippers/machines introduced The paper assumes the most simplified manipulation system (all objects are free flying)

Assembly Tree

Criticality based decomposition The space is divided based on criticalities of the translational freedoms of part pairs This is done using the notion of a “Non-Directional Blocking Graph”

Directional Blocking Graph of a Simple Assembly (translation only) AB CD B D A CD B D A CD

Non Directional Blocking Graph Regions are determined by comparing the local freedoms of two parts. Every such region is regular : in the sense that the Directional Blocking graph remains constant when d varies over it. Denoted as G(R,A) Unit Circle S1 represents directions BA CD Each region has a DBG associated with it.

The NDGB We can look at the complete NDGB of an assembly as an implicit representation of the set of all possible assembly algorithms. However, these algorithms may or may not be correct given accessibility constraints

Work backwards from assembled state when generating NDGB Goal state generally more constrained than initial configuration.... contacts in goal can be used to filter out impossible motions

Generating Algorithms NDGB is used to determine candidate algorithms and candidate portions The set of all candidate portions is computed in O(r^2u) time where r is the number of pairs of parts and u is the number of candidate partitioning Generating a candidate algorithm takes O(r^2n) time where n is the number of parts in an assembly.

Calculating DBG for Adjacent Regular Regions ● Simple way to optimize calculating adjacent regions. ● Start by computing G(R1) ● To generate G(R2) Take each contact edge in G(R1) that is parallel to D (E k ). (where D is the diameter of S1 that ends at R2) ● If the inner product of any direction in R1 and the outgoing normal to P i in E k is strictly positive then retract 1 from the weight of the arc connecting P i to P j

Visual A B S1: R1 R2 Normal dot D is positive for all D in R1 DGB(R1): DGB(R2): AB AB Note : If R1 is a singleton, then we ADD rather than Subtract weights R1 R2

Problems with using only infinitesimal translations ● Doesn't guarantee accessibility ● Doesn't allow rotational motions  There may be times when a translation is not possible but a rotation is: Assembly cannot be decomposed into translations. The set of translational DBGs cannot be represented discretely

Infinitesimal generalized motions ● Extension from 2d to 3d: Partition S2 rather than S1 (translation in 3d) ● To extend this to general motion we must use 6D vectors. Regions are now partitions of S5. S1 S2 S5 (ish?)

Generalized motion ● dX = (dx,dy,dz,da,dβ,dץ( ● Motion of Vertex V of P i is J v dX Where J v Is the Jacobian of transform that gives coordinates of V in the frame of P j ● N F J v dX ≥ 0 if the face does not intersect. ● N F J v dX = 0 defines a 5D hyperplane in 6D space. These planes define the boundaries of our regions. NFNF JvdX A1 A2 A3 A4 A5 A6 B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 C5 C6 dX JvdX In this case N F J v dX<0

Addressing Accessibility ● A new variant of the NDGB : Analyzes interferences for a family of non-infinitesimal motions (infinite translation) ● Two parts A,B (may or not be touching)... B blocks the “infinite translation” of A if the swept volume of A along d ● Important because the swept volume guarantees accessibility, while infinite translation does not. ● Set of direction where A is blocked is where Minkowski difference of A,B blocks origin. B A A A A A A A A A A A A A A A A A A A A A B D

Further Extensions ● Each NDGB is defined for a limited family of motions and describes only the class of assembly algorithms for these motions. ● One can imagine other NDBGs (for instance moving K+1 assemblies together simultaneously) ● This can be represented by the partitioning of S^3m-1 (3 directions for translation)

Computation time ● A singleton occurs only when 5 hyperplanes intersect, this means the arrangement contains O(K^5) regions where K is number of vertices ● Constructed in O(K^5) time using a mutli- dimensional topological sweep (?) ● DBG build in O(K) time, Crossing rule can be used to make NDBG in O(RK^5) time. Where is R is O(N^2) (number of pairs of contacts)

Results from Testing Not good Good

Assess Complexity

Complexity ● Admissibility : Admits at least one correct assembly algorithm ● Monotonicity : Each of its instructions merges a moved set into a complete subassembly ● P-handed: require P + 1 hands (includes one stabilizing hand) ● Prismatic : Extended translation ● Linearizable ● Minimal Length Algorithm ● Stack product: 1-prismatic, 1-handed, monotonic product that admits an algorithm where all instructions specify translation in the same direction. ● Number of Fingers

Number of Fingers Fingers are used to grasp and fix objects within a set from moving relative to each other Form closure if no element in the grasped set can move in any direction Form closure of order q prevents up to q+1 parts from moving.

Conclusion Automatically generate assembly algorithms Construct NDBGs by analyzing blocking relationships Determine the complexity of the algorithm to assess feasibility

The End!