1. Jeff AdamsADCATS 2000, BYU Feature Based Analysis of Selective Limited Motion in Assemblies Jeff Adams ADCATS 2000 Brigham Young University

2. Jeff AdamsADCATS 2000, BYU Project Motivation Need computational methods to support “Top-Down” assembly design philosophy Desire to take advantage of adjusting the position of parts during assembly to reduce variation. Methods are needed to calculate the location, direction, and amount of possible adjustment in an assembly. Locational overconstraint of compliant parts can lead to stored energy in the assembly. Methods are needed to detect the location and directions of overconstraint.

3. Jeff AdamsADCATS 2000, BYU Topics Top down design philosophy Overview of Motion Limit Analysis (MLA) Sketch of screw theory Application of screw theory to constraint analysis Examples

4. Jeff AdamsADCATS 2000, BYU Top Down Assembly Design Establish key characteristics (KCs) Construct Datum Flow Chain (DFC) Choose assembly features to physically realize the DFC Use MLA to calculate areas of adjustment and overconstraint Use control theory to optimize assembly sequence, location and amount of adjustment. Moving toward an analysis that suggests an assembly feature design

5. Jeff AdamsADCATS 2000, BYU Assembly Design Overview Assembly design theory has three elements –constraint defines how parts are located with respect to each other –assembly features on parts define where parts are located with respect to each other –tolerances on feature size and location define how accurately parts are located with respect to each other The Datum Flow Chain creates a top-down model that supports all three elements

6. Jeff AdamsADCATS 2000, BYU A DFC is a directed acyclic graph that defines the nominal relationships between assembled parts as well as assembly fixtures, tooling, and equipment such as robots A DFC characterizes a nominal design by identifying the part mates that convey dimensional control and by identifying the hierarchy that determines which parts or fixtures define the locations of which other parts DFCs also contain information on the type of mating feature and the type of motion constraint applied by that feature Datum Flow Chain (DFC) X,  z (6) Y, Z,  x,  y A B C

7. Jeff AdamsADCATS 2000, BYU Motion Limit Analysis (MLA) uses the mathematics of screw theory to model the ability of mechanical assembly features to allows or constrain rigid body motions in six degrees of freedom. If rigid body motion is allowed, the direction and quantitative amount of motion will be calculated. The ability to calculate rigid body motions of a part is important for enabling in-process adjustment during assembly to precisely establish key assembly dimensions. Motion Limit Analysis Overview

8. Jeff AdamsADCATS 2000, BYU Sketch of Screw Theory Chasle’s Theorem: Any motion of a rigid body can be reproduced as a rotation of the body about a unique line in space and a translation along that same line. T = [  x  y  z v x v y v z ]  angular velocity v =  x r  A  T r  d T v   x r A = rotation matrix, d = displacement vector

9. Jeff AdamsADCATS 2000, BYU Motion limits are defined by a vector. A Motion Limit Vector (MLV) is a 6x1 vector that describes the three numerical limits on translational motion in three independent directions, and the three numerical limits on rotational motion about axes aligned with the same independent directions. MLVs are defined for each assembly feature (inherent property for positive and negative motions). The main purpose of MLA is to combine the effects of several sets of MLVs associated with the features that are being used to connect one part to others, and calculate the net pair of MLVs that describe the resultant motion properties of the part as a whole. Motion Limit Vector

10. Jeff AdamsADCATS 2000, BYU Examples of Mating Features 18 and counting...

11. Jeff AdamsADCATS 2000, BYU Use of Screw Theory to Check Mobility and Constraint Create library of elementary features Each such feature has a twist matrix representation –Don’t need geometry! –Each row in the twist matrix represents an unconstrained degree of freedom Make constraining joints between parts by using one or more elementary features in combination Combine effects of all features and check degree of mobility and constraint using twist intersection algorithm by Konkar

12. Jeff AdamsADCATS 2000, BYU Combination of Mating Features in an Assembly Questions: Are there are any motions between parts? and, if any, What kind of motions? What is the quantitative amount of each motion?

13. Jeff AdamsADCATS 2000, BYU Motion Limit Analysis Overview N mating features N twist matrices 2N MLVs Resultant Twist matrix Ttot Find the max allowed rotation/translation motions Each  max is broken in  x  y  z components For each RDoF, and then for each ISA... Rotation allowed by each feature:  i Max rotation around ISA  max = min(  i) Resultant MLVs for each part Rotational DoFs Translational DoFs Analysis of Search for the max possible rotation axis Rotational DoFs Find the max rotation/translation components in the PCF

14. Jeff AdamsADCATS 2000, BYU Screw Theory for Constraint Analysis Poinsot’s Principle: Any set of forces and couples applied to a body can be reduced to a single force acting along a specific line in space, and a pure couple acting in a plane perpendicular to that line. A wrench is a screw describing the resultant force and moment of a force system acting on a rigid body. W = [f x f y f z m x m y m z ] f =  F i i = 1, …, n M i = r i  F i i = 1, …, n m =  M i i = 1, …, n r

15. Jeff AdamsADCATS 2000, BYU Analysis of Constraints Each twist has a reciprocal called a wrench expressed in part center coordinates as [Fx Fy Fz Mx My Mz] It represents all the forces and torques that the feature can transmit to a mating part Where motion is allowed, no force or torque can be transmitted, and vice versa The intersection of all wrenches acting on a part shows the amount of constraint on the part provided by those features If a part is constrained in some direction, then every feature can provide that constraint

16. Jeff AdamsADCATS 2000, BYU Analysis of a Combined Feature Made from Engineering Features Analysis results : Motion about Z is possible The rotation center is about f1 The amount of rotation can be calculated if peg and slot dimensions are known and slight clearance is assumed Mathematical Result T = [0 0 1 2 -2 0] Rotation occurs about Z axix at point [2,2,0]

17. Jeff AdamsADCATS 2000, BYU Constraint Analysis Results Overconstraint of a displacement in Y and a rotation about Z Overconstraint of displacement in Z Overconstraint of rotation about X Overconstraint of rotation about Y An error in the distance along the Y axis between the two features could cause difficulty in assembling the parts. The other three constraints come about because both features contain a planar mate.

18. Jeff AdamsADCATS 2000, BYU Second Example Analysis results : No motion is possible, assembly is fully constrained Overconstraint exists about X and Y Overconstraint exists along Z Mathematical Result: Twistmatrix is null W = [0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0]

19. Jeff AdamsADCATS 2000, BYU Assembly Using Features on the Parts. ASSEMBLY LEVEL DATUMS PART LEVEL DATUMS FORWARD SKIN AFT SKIN SPLICE STRINGER PLUS CHORD MATING FEATURE (SLOT) MATING FEATURE (HOLE)

20. Jeff AdamsADCATS 2000, BYU Other Uses for Screw Theory Evaluating local constraints during assembly sequence analysis Finding unstable subassemblies Determining if a DFC constrains a KC Determining how much adjustability there is in a Type-2 assembly Determining if KCs conflict

21. Jeff AdamsADCATS 2000, BYU Conclusions and Future Work A sufficient set of assembly features has been modeled to provide mathematical models of connections between parts in an assembly. Motion Limit Analysis has the ability to combine the motion characteristics of several assembly features, to yield two motion limit vectors describing the motion of the part as a whole. Work is being done to apply MLA to compliant parts using a simple FEA model. Combining MLA with other algorithms such as assembly sequence analysis, DFC analysis, control theory based tolerance analysis, and a CAD system has the potential to yield a comprehensive tool to support the “Top Down” approach to assembly design.

22. Jeff AdamsADCATS 2000, BYU Additional References J. D. Adams, "Feature Based Analysis of Selective Limited Motion in Assemblies," M.S. Thesis, Mechanical Engineering. Cambridge: MIT, February 1998. R. Mantripragada, “Assembly Oriented Design: Concepts, Algorithms and Computational Tools,” in Ph.D. Thesis, Mechanical Engineering. Cambridge: MIT, 1998. R. Konkar and M. Cutkosky, “Incremental Kinematic Analysis of Mechanisms,” Journal of Mechanical Design, Vol. 117, December 1995, pp. 589-596. R. Mantripragada and D. Whitney, “Modeling and Controlling Variation Propagation in Mechanical Assemblies Using State Transition Models,” IEEE Transactions on Robotics and Automation, vol 15, no 1, Feb. 1999. R. Mantripragada and D. Whitney, “The Datum Flow Chain: A Systematic Approach to Assembly Design and Modeling,” Research in Engineering Design, vol 10, 1998, pp 150-165. Whitney, D.E., R. Mantripragada, J.D. Adams, and T.W. Cunningham, “Use of Screw Theory to Detect Multiple Conflicting Key Characteristics,” ASME Design Engineering Technical Conferences, Las Vegas, Sept. 1999. Whitney, D.E., R. Mantripragada, J.D. Adams, and S.J. Rhee, “Designing Assemblies,” Research in Engineering Design, (1999) 11:229-253..