1

The Influence of Motion Paths and AssemblySequences on the Stability of Assemblies

Sourav Rakshit and Srinivas Akella, Member, IEEE

Abstract—In this paper we present an approach for thestability analysis of mechanical part disassembly considering partmotion in the presence of physical forces such as gravity andfriction. Our approach uses linear complementarity to analyzestability as parts are moved out of the assembly. As each partis removed from the assembly along a specified path duringdisassembly, we compute the contact forces between parts inthe remaining assembly; positive contact forces throughout thedisassembly process imply the disassembly sequence is stable(since the parts remain in contact with one another). However,if the part that is being taken out induces motion of otherparts in the remaining subassembly, we conclude the disassemblysequence is unstable. Thus we are able to simulate the entiredisassembly considering physical forces and part motion, whichhas not previously been done. We then show the influence ofpart motion on stable disassembly. In contrast to prior workon disassembly that has focused either on planning part motionsbased on only geometric constraints, or on analyzing the stabilityof an assembly without considering part motions, we explorethe relation between part motion and the selection of stabledisassembly sequences in 2D and 3D. We establish conditions thatcharacterize path-dependent assemblies, where motion paths canplay a significant role in stable disassembly. Since we track themotion of all parts in an assembly, instability inducing motionscan be identified and prevented by introducing appropriatefixtures, by selecting alternative disassembly sequences, or bychanging the motion paths. We extend the stability analysisfor single part disassembly to stability analysis of subassemblydisassembly. We additionally show that in the presence of friction,assembly and disassembly can be noninvertible.

Note to Practitioners—Maintaining the stability of an as-sembly as assembly or disassembly proceeds is critical duringproduct assembly, repair, and maintenance. While fixturescan ensure stability, they add cost and restrict access forparts and tools. We show that the stability of the assemblycan depend on the paths taken by the parts, and present anapproach for selecting paths that ensure stability from amongthe geometrically feasible paths. The main application is inassembly and disassembly planning, for reducing the fixturingrequirements and in planning assembly motions that do notcause instability. The results will also be useful in designingdisassembly operations during maintenance or repair, when itmay not be possible to fixture all parts. The stability analysiscan also be used by assistive robots in domestic environmentsfor tasks like stacking objects, and for safe removal of col-lapsed structures after disasters such as earthquakes.

This work was supported in part by NSF Awards CCF-0729161, IIS-1019160, and IIP-1266162.

S. Rakshit is with the Department of Mechanical Engineering, IIT Madras,Chennai 600036, India (Email: srakshit@uncc.edu) and S. Akella is with theDepartment of Computer Science, UNC Charlotte, Charlotte, NC 28223, USA.

The corresponding author is Srinivas Akella, Department of ComputerScience, University of North Carolina at Charlotte, Charlotte, NC 28223,USA. Tel: (704) 687-8573, Email: s.akella@ieee.org.

Index Terms—Assembly planning, stability analysis, motionpath, assembly sequence.

I. INTRODUCTION

Product assembly is a labor intensive and time consumingprocess [35, 8]. Automated robotic assembly is aimed at reduc-ing human cost and time. There has been significant researchon several aspects of assembly automation including the de-velopment of algorithms for assembly sequencing consideringpart geometry [36], grasping of objects considering physicalforces [21, 26], stability of subassemblies [23, 22, 24], andmotion planning for part removal [32, 14, 19]. The simplestassembly problem is one in which there are only two hands,and the assembly sequence is monotone, i.e., each part ismoved directly to its final position in the product withoutbeing placed in intermediate positions [36]. In this workwe consider monotone assembly of rigid parts using twohands. A classical strategy for assembly planning is assembly-by-disassembly [17]. This strategy is popular in assemblyplanning since a product in its assembled state has many moreconstraints than in its disassembled state; these constraintsreduce the search space for a planner. When the parts arerigid and when only geometric constraints are considered, aninverted disassembly sequence leads to a feasible assemblysequence. Thus the terms assembly and disassembly havebeen used interchangeably in the literature [36]. Note thatdisassembly by itself is important in repair, maintenance andend-of-life processing [18].

We follow a simple definition of stability: A disassemblyoperation is said to be stable, if apart from the part(s)being removed, all other parts in the assembly stay in theirrespective locations. We make the initial assumption that astable disassembly sequence when inverted will become astable assembly sequence. In our work the fixed support isconsidered as a hand, so there is only one moving hand. Thestability analysis we perform also has potential applicationsfor robots in domestic environments, where tasks like stackingbooks, dishes, boxes, and even blocks (e.g., in the gameJenga [34]) are common. It may also be useful for autonomousconstruction, and safe removal of collapsed structures duringrescue operations after disasters such as earthquakes.

The focus of our work is the stability of the assembly inthe presence of physical forces as each part is taken out of theassembly. We calculate the forces arising between the partsusing linear complementarity [2, 31]. The part motions are se-lected using the non-directional blocking graph (NDBG) [36].Our stability analysis is based on calculating the contact forcesand identifying relative part motion. The initiation of breaking

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of contact in frictionless assemblies is reflected as a positiveacceleration at the contact points, which is complementary tothe contact force at those points. For assemblies with friction,the relative motion is indicated by the sliding velocity. Wesimulate the motion of part(s) during disassembly, calculatethe contact forces and their complementary motions, anddetermine stability. Although the stability of assemblies hasbeen analyzed previously [6, 23, 22, 24], and such analysis hasbeen used to analyze the disassembly tree [25], the simulationof disassembly considering both physical forces and motionhas not been previously addressed. Thus our work can beincorporated into disassembly simulation and disassembly mo-tion planning based on physical forces and constraints. Further,we demonstrate that the stability of disassembly sequencesdepends not only on the disassembly sequence, but also onthe motion paths taken by the parts. We identify conditionsthat characterize such path-dependent assemblies. We showthat for path-independent disassemblies, it is sufficient toperform stability analysis at only the nodes of the AND/ORassembly graph. We extend the stability analysis for single partdisassembly to stability analysis of subassembly disassembly.Finally, we show that in the presence of friction, assembly anddisassembly can be noninvertible.

II. RELATED WORK

There are two main areas of research related to our work:stability analysis, and geometric analysis of assemblies. How-ever there has not been much prior work uniting these twoareas for assembly planning.

A. Stability Analysis

Blum et al. [6] first analyzed the stability of rigid blocks.They defined an assembly of rigid blocks as stable if and onlyif all the compressive contact forces between the blocks arepositive, and devised an algorithm similar to linear program-ming for solving the set of contact forces in the assembly.Palmer [27] investigated the computational complexity of sta-bility of polygons and gave definitions of guaranteed stability,potential instability, and infinitesimal stability. Boneschanscheret al. [7] extended the linear programming based approachof stability analysis to include insertion forces. Romney [30]presented a method for planar assemblies to concurrentlygenerate both an assembly sequence and a fixture to holdintermediate subassemblies. In this sequence/fixture co-designproblem, the fixtures must stabilize the (sub)assemblies againstgravity and the insertion forces, while permitting insertionpaths for the parts. Mattikalli et al. [23, 22] developed linearprogramming based approaches for identifying the set oforientations under which an assembly is stable in the presenceof gravity, both with and without friction. Mosemann et al. [24]developed a method similar to Mattikalli et al. and used itto analyze the stability of the subassemblies at each node ofthe AND/OR assembly graph [25]. Wolter and Trinkle [37]used a contact force based linear programming formulation tofind optimal location of fixels in an assembly. Goldberg andMoradi [15] developed vertical assembly plans for assemblywithout fixtures, with a minimalist motivation similar to ours.

With the exception of [25], very little work has used stabilityanalysis systematically for the disassembly process. Moreover,it focuses mainly on the static analysis of the disassembly tree.

The notion of a stable equilibrium is closely related toforce closure and form closure in robotic grasping [21, 26]and fixture design for a static assembly [9, 10]. Trinkle [33]formulated a first order stability criterion for grasped objectsbased on linear programming. An assembly is first orderstable if all kinematically possible virtual displacements of thesystem result in a strict positive increase in potential energyof the system. First order stability is equivalent to robustdirectional equilibrium for a system of rigid bodies and theirfixtures, as formulated by Baraff et al. [3]. Our definition ofstability is equivalent to directional equilibrium of Baraff etal. [3] and weak stability of Pang and Trinkle [28], which aredefined to hold for an assembly if the contact forces that arisein response to an external force exactly balance the externalforce and induce zero body accelerations.

Stable transport of assemblies of parts without grasping hasbeen studied by Bernheisel and Lynch [4, 5], who analyzedthe stable transport of planar arrangements of parts by pushingmotions. They identify force balance conditions that guaranteean assembly of parts stays assembled as it is pushed.

B. Geometric Analysis and PlanningFor a concise overview of geometric assembly planning

and sequencing, see [16]. The AND/OR graph representationof assembly sequences developed by de Mello and Sander-son [12] enables enumeration of all possible assembly se-quences for automated planning. The non-directional blockinggraph (NDBG) is a subdivision of the space of all allowablemotions of separation into a finite number of cells such thatwithin each cell the set of blocking relations between all pairsof parts remains fixed [16]. The NDBG representation uses thegeometry of the parts in the assembly to efficiently determinethe set of feasible motion directions for a part, thus reducingthe combinatorial complexity of disassembly sequencing [36].The geometric information in the NDBG together with thesequencing information in an assembly AND/OR graph is thenused to perform assembly sequencing. Halperin et al. [17]developed a motion space approach to generalize the NDBGto more general part motions.

More recently, disassembly planning has been addressedas a motion planning problem in the composite configu-ration space of the individual parts by applying sampling-based motion planners for planning part removal paths fordisassembly [32]. Techniques have included sampling basedon the geometry of reachable configurations. An RRT-basediterative planning approach that gradually decreases the al-lowed amount of interpenetration has been used for single partdisassembly [14]. Le et al. [19] describe an RRT-based methodfor simultaneous disassembly sequencing and path planning; itcan handle nonmonotone disassembly sequences. While thesemotion planners generate paths that avoid part interference,they do not consider any physical constraints such as gravity,friction, etc.

Portions of the work in this paper previously appeared in[29].

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III. PROBLEM AND APPROACH

A. Motion Stability Problem

Given a geometrically feasible disassembly sequence and amotion path for an assembly, the motion stability problem isto identify whether the disassembly sequence and motion pathare stable.

To solve the motion stability problem, we consider eachpart as it is removed from the assembly along a specifiedpath, and perform stability analysis to check whether thestatic equilibrium conditions are satisfied along the path forall parts of the assembly other than the part being movedout. We declare a disassembly sequence and motion pathto be stable when there is no relative motion between parts(excluding the part being moved by the gripper) along the path.Using this motion stability analysis, we can identify stablesequences and motion paths for disassembly from a given setof geometrically feasible disassembly sequences and motionpaths for an assembly.

We make the following assumptions. We assume the assem-bly consists of polygonal parts (in 2D) or polyhedral parts (in3D). We assume that each part that is moved out is held bya gripper. The moving part is assumed to undergo quasistaticmotion with finite translations. There is perfect position controlof the gripper and hence of the part that it grasps. The gripperdoes work only on the part that it holds and moves, exceptwhen moving a subassembly. The weight of the moving partis supported by the gripper, unless otherwise stated. When thegripper does not support the part, we assume it applies forcesonly along the motion path. Following the convention in theassembly planning literature, we do not consider collisions ofthe gripper with the assembly. We assume frictionless contactunless otherwise stated.

We use linear complementarity [11] to detect whether andwhen the assembly becomes unstable as each part is removedfrom the assembly. If linear complementarity finds a solution,i.e., all the contact forces are non-negative and there is no rela-tive motion between parts (excluding the part being moved bythe gripper), then the assembly state is considered stable. Thelinear complementarity problem (LCP) proposed by Baraff [2]for analysis of frictionless cases calculates the relative move-ment in terms of the relative acceleration between the parts inthe assembly. According to Baraff’s model, for every contactpoint j in the assembly, the normal contact acceleration aj andthe normal contact force Fj form a complementary pair, i.e.,Fj ≥ 0, aj ≥ 0, and Fjaj = 0. This statement, whichimplies that when Fj > 0, aj = 0, and when aj > 0, Fj = 0,is represented succintly as 0 ≤ Fj ⊥ aj ≥ 0. We can sum upthe complementarity conditions for all n contact points, andsince aj and Fj are linearly related [1],

n∑j=1

Fjaj = FTa = FT (AF + b) = 0 (1)

where A ∈ Rn×n is a positive semidefinite matrix. Allexternal and body forces are grouped in the vector b ∈ Rn.

If friction is included, then in addition to normal forces andaccelerations there will be a tangential component of force and

acceleration at every contact point. When the static frictionforce limit is reached at any contact point, the matrix A isno longer positive semidefinite. Baraff’s method cannot solveproblems when A is not positive semidefinite. The Stewart-Trinkle model [31] gives a complementarity formulation formultibody contact problems with dynamic friction that canbe solved using Lemke’s algorithm [11]. Hence, we use theStewart-Trinkle model for disassembly problems with friction.In the Stewart-Trinkle model, at each contact point j, thecomplementary constraints are:

0 ≤ dnj ⊥ pnj ≥ 0 (2)

0 ≤ ρj = λjej + WTfjv ⊥ pfj ≥ 0 (3)

0 ≤ ζj = µpnj− eTj pfj ⊥ λj ≥ 0 (4)

where dnjis the normal distance between the two parts at j,

pnj is the normal component of the contact impulse, pfj isthe friction component of the contact impulse, ej ∈ RD isa vector of ones, D is the number of edges of the frictionpyramid, λj is a variable that approximates the magnitudeof the sliding velocity, µ is the Coulomb friction coeffi-cient, nd is the dimension of the configuration space, andWfj ∈ Rnd×D is the friction contact wrench matrix thattransforms the generalized velocity v ∈ Rnd of the part alongthe tangential component of the friction cone. For a two-dimensional problem (nd = 3), at each contact point j, thereare only two directions {tj ,−tj} of the friction componentof the contact impulse, each perpendicular to the outwardcontact normal cnj . The friction contact wrench matrix Wfj

is[

tj −tjrj ⊗ tj −rj ⊗ tj

], where rj is the vector from the

part centroid to contact point j, and ⊗ is the two-dimensionalequivalent of cross-product.

Using the Stewart-Trinkle model, we formulate the motionpath stability problem as a linear complementarity problem:

0 ≤

Ann Anf 0

Afn Aff E

µ −ET 0

ptn

ptf

λt

+

btn

btf

0

=

dtn

∆t

ρt

ζt

⊥pt

n

ptf

λt

≥ 0

(5)where superscript t indicates value at time t. pn =

[pn1, . . . , pnK

]T , pf = [pf1 , . . . ,pfK ]

T , and λ =

[λ1, . . . , λK ]T are the concatenated normal contact impulses,

friction impulses, and λs respectively for all the K contactpoints in the assembly, and Ann = WT

nM−1Wn,

Anf = WTnM

−1Wf ,Aff = WT

f M−1Wf ,

Afn = WTf M

−1Wn,

bf = WTf M

−1(Fb + Fext

)∆t and

bn = WTn

(M−1

(Fb + Fext

)∆t+ vt−1

)+ dt−1

n /∆t.

M is the concatenated mass matrix, Wn = [Wn1 , . . . ,WnK]

is the concatenated normal contact wrench matrix whereWT

nj=[cnj

rj ⊗ cnj

], Wf = [Wf1 , . . . ,WfK ] is the

concatenated friction wrench matrix, µ = diag [µ1, . . . , µK ],E = diag [e1, . . . , eK ], dn = [dn1

, . . . , dnK]T , ρ =

[ρ1, . . . ,ρK ]T , ζ = [ζ1, . . . , ζK ]

T , vt is the concatenated

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generalized velocity at time t, and Fb and Fext are theconcatenated body and external forces for all the parts.

After each time step ∆t, we update the position of themoving part, and then use linear complementarity (Equation 1or Equation 5) to calculate the contact forces and any relativemotion (acceleration or sliding velocity) of the parts in theassembly. Tangential accelerations need to be checked for onlythose bodies in the assembly that can move according to theNDBG of the assembly, ignoring bodies that are blocked inthe tangential direction. After obtaining the contact forces, wecan check the existence of tangential acceleration in a bodyby calculating the net force in the tangential direction.

B. Simulation and Stability Analysis of Disassembly

We now describe the algorithm to perform stability analysisat each stage of disassembly; it uses linear complementarity forcontact force computation, and a path planner to identify mo-tion paths for the parts that are consistent with the NDBG. Wedescribe the approach for evaluating a disassembly sequencefor an assembly (see Algorithm 1).

Let AN (qN ) be a complete assembly of N parts withconfiguration qN , S =<P1, P2, . . . , PN> be a disassemblysequence with parts numbered in their disassembly order,and AN−i(qN−i) be the remaining assembly after the first iparts of the sequence have been removed. Given part Pi+1 atits current configuration q, the function MotionPath returnsa geometrically feasible path <d> as a sequence of equalmagnitude displacement vectors dk for Pi+1 to move out ofAN−i(qN−i). Each motion step dk ∈ <d> is executed andthe part Pi+1 is moved for time ∆t with a unit velocityalong dk. After Pi+1 has moved to its updated position q, thefunction LCPsolve(AN−i(qN−i), Pi+1(q)) is called to solvethe LCP problem of Equation 1 (for the frictionless case) orEquation 5 (for the friction case) and compute the contactforces. LCPsolve returns true (i.e., stable) if none of thecontact relative motions (accelerations for the frictionless caseand velocities for the friction case) are greater than zero. If anyof the contact relative motions are greater than zero, LCPsolvereturns false (i.e., unstable) and the simulation is terminated.

IV. EXAMPLES

We now present a few examples to illustrate the approach.For a specified disassembly sequence, the parts in the assemblyare moved along motion paths with directions consistent withthe NDBG. At each time step, we calculate the contact forcesbetween different bodies by linear complementarity. If at anytime during the simulation, the part that is being moved outby the gripper induces a relative motion at any contact pointin the rest of the assembly, we conclude that the assembly isunstable. Unless otherwise stated the gravitational accelerationg is assumed to be unity, mass density of the blocks is unity,and the simulation time step is 1 second. We will refer to apoint i as vertex i if it is a vertex of the body or as a contactpoint i if it is the corresponding point on the mating body. Wewill depict the block moved by the gripper in red in the figures.For stability analysis, we use our MATLAB implementation ofBaraff’s complementarity algorithm for the frictionless case,

Algorithm 1 Stability Analysis of a Disassembly SequenceInput: Assembly AN , Disassembly SequenceS =<P1, . . . , PN>

1: for i = 0, . . . , N − 1 do2: <d>= MotionPath(AN−i(qN−i), Pi+1(q))3: if <d>== ∅ then4: return GEOMETRICALLY INFEASIBLE5: end if6: for k = 1, . . . , length(<d>) do7: vgripper = dk/‖dk‖8: q ← q + vgripper ·∆t9: Stableflag = LCPsolve(AN−i(qN−i), Pi+1(q))

10: if Stableflag == FALSE then11: return UNSTABLE12: end if13: end for14: end for15: return STABLE

unless otherwise stated, and for the case with friction, we usea MATLAB interface to the LCPPATH solver [13].

A. Simple Example with Blocks

This example (Figure 1) has a set of frictionless blocksarranged such that if block C is lifted before block D,block B topples. We use Baraff’s complementarity methodto analyze this. Consider the disassembly sequence C-D-B-A,with vertical motion of C (any direction in the upper halfplaneabove B is permitted by the NDBG). As C is lifted up, we plotthe contact forces, normalized with respect to the weight ofmoving block C, in Figure 2. At 0 s all the contact forces arepositive. However, at 1 s when block C is detached from therest of the assembly, acceleration at contact point 4 is positiveindicating that the rest of the assembly has become unstable.A similar situation occurs when block C is moved horizontallyto the left or right. Thus, all disassembly sequences startingwith the removal of block C are unstable.

0 10 20 30 40 50 60−5

0

5

10

15

20

25

30

35

40

E

A g

1 2

4 3

10

11

5

B 13 14 7

6

16 15

C D

12

89

Fig. 1: A simple blocks example: An arrangement of blocksin an equilibrium configuration. The blocks are identified byalphabets, and vertices by numbers. Block E is the fixed table.Based on [6].

By similar analyses, we can show that all disassemblysequences that start with the removal of blocks A, B, or Care unstable. Now consider the sequence D-C-B-A. We show

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Fig. 2: Plot of contact forces (normalized by weight of themoving block C) against time as block C is lifted up. As soonas C is lifted up, contact point 4 has positive accelerationindicating that the remaining assembly is unstable.

Fig. 3: Plot of contact forces for disassembly sequence D-C-B-A. X-axis is time, Y-axis is contact force normalized by theweight of block B. The forces at the contact points of eachblock remain positive until that block is lifted. This impliesthat the disassembly sequence is stable.

the contact forces (normalized with respect to the weightof the block B) in Figure 3. As each block is lifted upvertically, the contact forces for that particular block becomezero. However all other contact forces are positive indicatingthe remaining subassemblies in all cases are stable. Fromthe complementarity formulation, this implies that the contactaccelerations are zero, i.e., none of the contacts are breaking.Hence, D-C-B-A is a stable disassembly sequence, and is infact the only stable disassembly sequence.

B. Example: Stable Disassembly with and without Fixels

When disassembly sequences selected using only geometricconstraints lead to instability, we can use fixels (i.e., pointfixtures) to constrain parts that become unstable as other partsare removed. Using stability analysis, the parts that becomeunstable at the time of removal of other parts can be identifiedand stabilized by fixels, or alternative disassembly sequencescan be generated.

0

10

20

30

40

50

60

D

612

g

8

C

15

4

A

3

211

7 9 10

B

Fig. 4: Blocks A, B, and C rest on a frictionless base D.

This example illustrates the use of fixels in maintainingstability of an assembly. Consider blocks A, B, and C on tableD (Figure 4). Only block A can move vertically up accordingto the NDBG. However, as soon as A is lifted up, B becomesunstable (Figure 5 (a)), with rightward accelerations of vertices9 and 12. The instability in B when A is moved out can beprevented by application of fixels at appropriate locations onB (Figure 5 (b)). Fixel positioning can be optimized usingexisting methods, e.g., [37].

V. THE EFFECT OF MOTION PATHS ON STABILITY

We now examine the question: Are part motion pathsimportant during disassembly and assembly? The stabilityanalysis that we have presented so far can alternatively beperformed by analyzing the stability of the subassemblies ateach node of the assembly sequence AND/OR graph, as inMosemann et al. [25]. In our work, we simulate the process ofdisassembly by considering part motion. We next demonstratethat motion paths are also important for determining stabledisassembly sequences. We show that instability can occur notonly due to the order in which parts are removed, but also dueto the motion paths by which they are removed.

We will call the class of problems where stable disas-sembly depends on motion paths path-dependent disassemblyproblems. The complementary class of stable disassemblyproblems where motion paths do not affect stability will bereferred to as path-independent disassembly problems. Nextwe present both 2D and 3D examples to illustrate how stabledisassembly depends on the choice of the motion path.

A. Influence of Motion Path in 2D

Consider the task of disassembling blocks A, B, C, and Dfrom the part E on a fixed table (Figure 6). This example,where the blocks are constrained to move through the exitsI to V in Figure 6, illustrates the effect of motion paths onstability. A and B can move out through exit I, whereas C andD can move out through exits I, II, III, IV, and V. The shoulderson the rim of the cavity holding A and B prevent C and Dfrom contacting A when exiting through I. All the blocks arefrictionless. The weight of the moving part is not supportedby the gripper unless the part itself becomes unstable.

Since this example is symmetric about the vertical axis, weanalyze the effect of the motion of block C to illustrate theinfluence of motion paths (i.e., choice of exits) on the stabilityof the assembly. C has geometrically feasible paths out of theassembly through any of its three nearest exits I, II, and III.

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(a) (b)

Fig. 5: Plot of contact forces as block A is lifted up vertically. X-axis is time, Y-axis is contact force normalized by the weightof the moving block A. Note that since there can be more than one normal at a contact vertex, we indicate the bodies betweenwhich the contact forces act in the legend of forces and accelerations. (a) Without fixels: As soon as block A is lifted, blockB becomes unstable. This is indicated by positive accelerations at vertices 9 and 12 (scaled by a factor of 10 and shown inred). (b) With fixels: Since the preceding analysis indicates that B loses contact in the horizontal right direction, we put fixelson the right hand free face of B at vertex 11 and midway between vertices 11 and 10 (as shown in the bottom figures). Now,when A is lifted up, B stays in position, and none of the contact vertices have positive accelerations.

(a) (b)

Fig. 8: The interplay of assembly sequence and motion paths. Y-axis shows the contact force normalized by the weight of C,and acceleration, and X-axis is time. (a) Block A is first moved out through center exit I. C is then moved to the left. At 25 s,before C is in position to exit through II, the assembly becomes unstable. So when A is removed first, C cannot exit througheither exit II or III. (b) A is moved out first and then C is moved to exit I. In this case the assembly remains stable.

We first illustrate the influence of motion paths on stability.Consider the removal of C through exit III. As C is movedto the left through the passage leading to exits II and III, thecontact force at vertex 17 increases and that at 18 decreases(Figure 7) to counterbalance the increasing moment producedby the weight of C. At 25 s when C is directly below exit II, itcan be moved up towards exit II without instability. Howeverat 29 s when C is at exit III, the assembly becomes unstable.This is indicated by positive acceleration at vertex 18.

When C is lifted up and moved towards exit I, no instabilityis induced in the assembly. Hence, for disassembly sequencesthat begin with the removal of C, the first step in the disas-sembly sequence is stable when C exits through I and II, andnot through exit III. A similar situation occurs with D.

Next we illustrate the interplay of motion paths and assem-

bly sequences in maintaining stability. Consider removal ofA first, followed by C. At 25 s when C is positioned to exitthrough II (Figure 8 (a)), the assembly becomes unstable dueto positive acceleration at vertex 18. Hence C cannot exit eitherthrough II or III when A has been moved out. When C is liftedup and moved to the right towards exit I (Figure 8 (b)), C canbe moved out through I at 18 s by the gripper (lower righthand figure of the assembly in Figure 8 (b)). Thus when A isabsent, C can be moved out stably only through exit I.

A similar analysis shows that removal of C after both A andB have been removed always leads to instability. We summa-rize the stability analysis results for sequence and motion pathdependence in Table I. The motion path dependence for thedisassembly of D is symmetric to that of C.

7

(a) (b)

Fig. 9: (a) An assembly consisting of frictionless blocks A, B, and C. (b) Wireframe view of A, B, and C showing the contactsurfaces. The contact polygons between the blocks where contact forces occur are shown using different colors and numberedwith Roman numerals. For example, the contact polygon between A and B is I with contact vertices 1, 2, 3, and 4.

0

10

20

30

40

50

A

Bg

16

D15

13

14

IVIII

VIII 232410

11

C12

9

E 4

1

3

2

7

65

8

17 18

Fig. 6: Example to illustrate the effect of motion paths onstability. A, B, C, D are movable frictionless blocks restingon a nonfixed body E, that rests on a table (colored grey).The exits are numbered using Roman numerals.

Sequence Path Stable UnstableA-B-C All XA-C-B (I, I, I) XA-C-B (I, II, I), (I, III, I) XC-A-B (I, I, I), (II, I, I), (III, I, I) X

TABLE I: Stability analysis results for different sequence andmotion path combinations for the assembly of Figure 6. Thepaths (indicated by exits) for parts A, B, and C are listed asordered triples in the order of the sequence.

B. Influence of Motion Path in 3D

We now consider a 3D example to show that the LCPcalculations for stability analysis can be extended to 3Dassemblies as well. Figure 9 (a) shows an assembly consistingof the frictionless blocks A, B, and C resting on a fixedsupport (table). Geometrically feasible disassembly paths existfor all the blocks. However, it is easy to see that if B istaken out first then both A and C will fall down. Hence,we only consider the disassembly sequences A-C-B and C-A-B for stability analysis. This example was solved using theLCPPATH solver [13].

When A is lifted up, the weight of C causes B to topple;

Fig. 7: Plot of normalized contact force with respect to theweight of C and acceleration versus time. When C is movedto the left and lifted up to exit through II at 25 s, no instabilityis induced. However, when block C is moved to the extremeleft towards exit III, at 29 s the rest of the assembly becomesunstable, as indicated by positive acceleration at vertex 18.

this is manifested by positive accelerations at vertices 9 and 12(Figure 10). Thus A-C-B is an unstable disassembly sequence.Next consider C-A-B. To illustrate the effect of motion path,we consider two paths for the removal of C, one along +X andthe other along the −Y direction in Figure 9 (a). In both cases,the gripper supports the weight of C to the extent necessary toprevent C from falling down. Figure 11 shows the normalizedcontact forces (with respect to the weight of C) when C ismoved along the −Y direction. Note that the contact forcesat 9 and 10 increase and at 11 and 12 decrease as C is movedin the −Y direction. At 12 s the assembly becomes unstableand we see positive accelerations at 11 and 12.

Next consider moving C in the +X direction. The supportpolygon 13-14-15-16 of the gripper prevents C from topplingas the downward projection of the centroid of C moves outsidethe contact polygon 5-6-7-8 (Figure 12 (b)). The contact forces

8

(a) (b)

Fig. 12: C is moved in +X direction. To prevent toppling of C, the front end of C is supported by the gripper. (a) C in anintermediate position with the support of the gripper G at the leading end. (b) The wireframe diagram of the blocks with thecontact polygons. The contact polygon between C and the gripper support is indicated by vertices 13, 14, 15, and 16.

0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s) →

F/M

g, a

cc →

F1,F2,F3,F4F5,F8F6,F7F9,F12F10,F11a9,a12

F1,F2,F3,F4

F6,F7

F10,F11

F9,F12F5,F8

a9,a12

Fig. 10: A is lifted up from the assembly of Figure 9. Thecontact forces are numbered according to the vertices of thecontacting surfaces in Figure 9 (b). X-axis represents time. Y-axis represents the normalized contact forces with respect tothe weight of A, and contact accelerations. When A is liftedup, positive accelerations at vertices 9 and 12 indicate theremaining assembly is unstable.

are shown in Figure 13. There are no positive accelerations atthe contact points, which implies stability. As C is moved inthe +X direction, the variation in load over B is manifestedby the decrease in the contact forces at 9 and 12 and theincrease in the contact forces at 10 and 11. The edge 13-16of the support bears a higher fraction of the weight of C thanthe edge 14-15. The load on B through the edge 5-8 staysconstant until the end (25 s) when edges 6-7 and 5-8 merge.

Thus we have shown using stability analysis that the dis-assembly sequence A-C-B is unstable under gravity, and thedisassembly sequence C-A-B is path dependent.

VI. CHARACTERIZATION OF MOTION PATH DEPENDENCE

To characterize motion path dependence, we establish con-ditions for stability and instability of frictionless disassembly

0 2 4 6 8 10 12

0

0.5

1

1.5

2

Time (s) →

F/M

g, a

cc →

F1,F2,F3,F4F5,F8F6,F7F12F10F11F9100*(a11,a12)

F10

F9

F11

F12 F5,F8

a11,a12

F1,F2,F3,F4,F6,F7

Fig. 11: Plot of the normalized contact forces and accelerationsversus time when C is moved horizontally in the −Y direction(Figure 9 (a)). The contact forces are normalized with respectto the weight of C. At 12 s the whole assembly becomesunstable and vertices 11 and 12 lose contact with the table.Accelerations at contact points 11 and 12 are shown magnifiedby a factor of 100. Top views of assembly shown at bottom.

problems. These provide a way to identify potential insta-bility even without performing a stability analysis along the(dis)assembly paths. The results below are stated in the contextof assembly stability as one part is removed. Let the partbeing grasped and moved by the gripper be P , the assemblyincluding P be A, and the remainder of the assembly withoutP be P , where P = A− P .

Proposition 1: If P is unstable, then all motion paths forthe disassembly of P from the assembly A are unstable.

Proof: All motion paths for disassembly of part P ulti-mately lead to the removal of P from assembly A. Irrespectiveof the motion path, the removal of P is equivalent to thedetachment of P from assembly A. Hence, if detachment ofa part from the assembly leads to instability, all motion pathswill lead to instability.

9

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9F

/Mg

→

Time (s) →

F1,F2,F3,F4F9,F12F10,F11F5,F8F6,F7F14,F15F13,F16

F1,F2,F3,F4

F9,F12

F10,F11

F14,F15

F13,F16

F5,F8

F6,F7

−10 0 10 20 30 40 50050−5

0

5

10

15

20

25

30

35

40

−10 0 10 20 30 40 50 60 70050−5

0

5

10

15

20

25

30

35

40

Fig. 13: Plot of the normalized contact forces versus time whenC is supported by the gripper and moved in +X direction. Thecontact forces are normalized with respect to the weight of C.The assembly stays stable throughout the motion path.

P

Fig. 14: Staircase of blocks supported by the block P underthe topmost block. When P is moved out in any geometricallyfeasible direction from under the staircase, the staircase willno longer be stable.

Figure 14 illustrates the proposition with an example.

Proposition 2: A sufficient condition for motion path inde-pendence in a disassembly problem is that the weight of themoving part is fully supported by the gripper.

Proof: Stability of an assembly depends on the contactforces between the mating parts. If the weight of part P is fullysupported by the gripper, then P will not exert any contactforce on P (under our assumptions in Section III-A). Thesituation is identical to the case when P is not part of A.Hence, motion paths cannot affect stability if the part’s weightis fully supported by the gripper.

The contrapositive of Proposition 2 is that if the disassemblyproblem has path dependence, the moving part P is not fullysupported by the gripper. Proposition 2 along with the fact thatthe gripper must fully support P when moving it vertically,leads to the following result.

Corollary 1: Part movement only in or opposite to thedirection of gravity will not cause assembly instability dueto the motion path.

We next identify conditions under which instability canoccur when a part P is moved while not being fully supportedby the gripper. (See Figure 15 and Figure 16 for stable andunstable examples.) We assume P is moving along a planarsurface in the direction m, and that the assembly A rests

P

dxC

C

xP

O

m

Z

X

Fig. 15: Stable example. Part P can move along its supportsurface without the assembly becoming unstable.

C

P

Z

X

m

v

O

xP

dxC

Fig. 16: Unstable example. Instability can occur when part Pmoves along a surface at a sufficiently large distance from asufficiently narrow base of support.

on a horizontal surface. We also assume that the contactpolygon has contiguous line contact (in the 2D case) or surfacecontact (in the 3D case) with the horizontal support surface.For polyhedral parts that are in contact with one another, thecontact polygon is the boundary of the region of contact.

Proposition 3: If the maximum distance xP between thecenter of mass of P as it moves on its supporting planar sur-face and the vertices of the contact polygon on the horizontaltable along the projected motion direction is sufficiently largerthan the maximum distance dCx between vertices of the contactpolygon on the horizontal table along the projected motiondirection, instability will occur.

Proof: Let the projection of the motion direction m onthe horizontal table be the projected motion direction v. Weproject the assembly on a plane V⊥ perpendicular to thehorizontal table and parallel to v. In 2D, v is parallel to theline representing the table, and V⊥ is the plane in which theassembly lies. Let the part in contact with the table be C. LetSC = {v1, v2, . . . , vK} be the set of vertices of the contactpolygon of C on the table. We locate the origin O at thevertex in SC farthest from the centroid of P along v, theX-axis along v, the Z-axis along the upward normal to thetable in the plane V⊥, and the Y -axis by the right hand rule.

10

So when P moves along m, the moment due to the contactforces will only change about the Y -axis.

Let mP be the mass of P and xP be the X-coordinate ofthe center of mass (COM) of P . Let mP be the mass of theremainder of the assembly P and xP be the X-coordinate ofthe COM of P . Let {xv1 , xv2 , . . . , xvK} be the X-coordinatesof {v1, v2, . . . , vK}. Let dCx = max{|xv1 |, |xv2 |, . . . , |xvK |}.Let Fz be the net force along the Z-direction and My be thenet moment along the Y -direction caused by the contact forcesfrom the horizontal table. Considering equilibrium of forcesalong the Z-direction and moments along the Y -direction,

mP g +mP g = Fz =∑i∈SC

F viz (6)

mP gxP +mP gxP = My =∑i∈SC

F viz xvi (7)

From Equation 7 we can write,

mP gxP

dCx+mP g

xP

dCx=My

dCx=∑i∈SC

F viz xvi

dCx≤∑i∈SC

F viz (8)

As P moves along m, mP gxP /dCx is constant while themagnitude of mP gxP /dCx increases; the magnitude of My

increases to maintain equilibrium. However, the maximumvalue that My/d

Cx can reach is Fz , which is constant for

the assembly. If the polygonal surface on which P moves issuch that the projected distance of its farthest vertex from theorigin O along the X-direction is much larger than dCx , thenP can move sufficiently far from O such that xP >> dCx , andEquation 8 is no longer satisfied. This condition will lead toinstability.

We now illustrate the above proposition with respect to thepresented 2D and 3D examples. In the 2D example (Figure 6),the horizontal projection of the base is the length of theedge between vertices 17 and 18, which is smaller than thehorizontal projection of the top surface over which C moves.Hence, the moment arm of C can be much larger than themoment arm of the contact forces from the horizontal table,i.e., the length of the edge 17-18. The instability (Figure 7)is induced by the increasing moment as C moves to the left.The increasing moment eventually makes the contact forcedecrease to zero at vertex 18, after which instability is induced.

In the 3D example, note that the horizontal projectionof the base along the motion direction along the negativeY -axis (the edge 10-11 in Figure 9), is smaller than thehorizontal projection of the top surface of B along the samedirection (edge 2-3). In this case, when C is moved in the−Y direction, the instability is induced due to the increasingdifference between contact forces at {9,10} and {11,12} tocounterbalance the moment due to the weight of C. Notethat if the weight of P is fully supported by the gripper,then mP gxP /dCx is zero in Equation 8, and the equationsof static equilibrium are satisfied for all paths, consistent withProposition 2.

We use linear complementarity to perform stability analysisat discrete configurations as parts are moved out of the

assembly. We ask the question: Given that at two discreteconfigurations the assembly is stable, when can we guaranteethat the assembly is stable for all configurations betweenthose two configurations? We prove that if disassembly is pathindependent as defined in Section V (for example, when thepart is supported by the gripper), the assembly will be stableat all configurations between a pair of stable start and endconfigurations.

Proposition 4: For path-independent disassembly, if theassembly A at any two configurations of a path is stable aspart P is moved, then it is stable at all configurations alongthe path between the two configurations.

Proof: Let i, j, and k be three different configurationsof the assembly A along a path for P , with k being anintermediate configuration between i and j as P is movedout. Let V q

P and V q

Pdenote the potential energy of P and

P respectively at a configuration q. Let i and j be stableconfigurations, and assume the intermediate configuration k isunstable. This implies that at configuration k, some part(s) inP will begin to move and gain kinetic energy. Since the gripperdoes work on only P by our assumptions, the gain in kineticenergy of part(s) in P will lead to a decrease in potentialenergy of P at k, i.e., V k

P. The parts at the configuration k

that are not in equilibrium will move to new equilibrium stateswith lower potential energy. Since all parts other than P are intheir unchanged locations at configurations i and j, thereforeV iP

= V j

P> V k

P. However the gripper does work only on P ,

and hence it cannot do work and increase the potential energyof the parts that have moved into lower potential energy states.Hence, for all configurations at and after k on the path, thepotential energy of P will be less than or equal to V k

P. Since j

lies after k, therefore, V kP≥ V j

P. But this directly contradicts

our assumption that V kP< V i

P= V j

P. Therefore k must be a

stable configuration.

Corollary 2: For the case of path-independent disassembly,checking for stability at the nodes of the AND/OR graph ofthe disassembly tree is sufficient to verify stable disassemblysequences.

Proof: For path-independent disassembly, it follows fromProposition 4 that if the two configurations that correspond tothe state of the assembly with and without the part P respec-tively are stable, then P will be stable at all configurationswhile P is being disassembled. Thus for the case of path-independent disassembly, checking stability at the nodes ofthe AND/OR graph of the disassembly tree is sufficient toverify stable disassembly sequences.

VII. DISASSEMBLY OF SUBASSEMBLIES

So far we have applied stability analysis to the case of singlepart disassembly. However, the efficiency of disassembly canbe increased by simultaneous disassembly of multiple parts assubassemblies. In some cases, disassembly of subassembliesmay be the only stable disassembly process. We present anexample to illustrate that stability analysis can also help inselection of stable subassemblies. Figure 17 shows four blocksA, B, C, and D forming an assembly. There is no friction

11

between the blocks, and between the blocks and the support.For two handed disassembly, the possible subassemblies fordisassembly are: {A-B, C-D}, {A-B-C, D}, {A-C-D, B}, and{A-C, B-D}. Removal of any single part from the assemblyresults in instability. A or C cannot be lifted up independentlywithout causing instability, and have to be held together andlifted simultaneously for stable disassembly. B and D can bemoved sideways, but the motion of either leads to instability.Figure 18 (a) shows the contact forces and the accelerationsleading to instability when B is moved to the left by thegripper. After B has moved to the left, at 10 s, A becomesunstable (shown by contact accelerations at vertices 7, 33 and38 in Figure 18 (a)).

On the other hand, when A is moved to the left, it pushes Balong with it, and the disassembly is stable. The contact forcesare shown in Figure 18 (b). A can then be stably disassembledfrom B, and similarly C can be stably disassembled fromD. Thus, the stable disassembly of the example shown inFigure 17 can only be executed by considering subassemblies.

0

10

20

30

40

50

2015

CABD

35

171413 22

1 2 34

38

7

37

33

3

Fig. 17: Example for disassembly of subassemblies. Friction-less blocks A, B, C, and D rest on a fixed support. Thevertices relevant for the contact forces are shown. As removalof individual parts leads to instability, {A-B, C-D} is the onlystable subassembly sequence for disassembly.

VIII. NONINVERTIBILITY OF ASSEMBLY ANDDISASSEMBLY

When only geometric constraints are considered, a two-handed monotone disassembly can be inverted for assem-bly [17]. However, when physical forces are considered, atwo-handed monotone assembly and disassembly might not beinvertible. Although this has been previously noted [20, 17],we are not aware of a prior exploration of this issue based onsimulation. We construct a simple example to illustrate howa physical force, i.e., friction, may lead to noninvertibility ofassembly and disassembly. The assembly consists of two rigidblocks A and B (Figure 19) that can be disassembled either bymoving vertically, or by moving horizontally. The disassemblysequence is A-B, and the assembly sequence is B-A. In thevertical direction, assembly and disassembly are invertible.

Let us now consider disassembly when A is moved outto the right. We select the coefficients of friction betweendifferent surfaces as shown in Figure 19. That is, µ is 0.2between A and B, 0.05 between B and its support surface,and zero between all other surfaces. The weight of each blockis one unit. The complementarity conditions for this systemare given by Equation 5, and we use the LCPPATH solver to

Fig. 19: A noninvertible disassembly example. The blocks canbe disassembled either by lifting from the top (invertibly), orby moving horizontally (noninvertibly). The friction coeffi-cients between different surfaces are shown.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (s) →

p/M

g∆t,v

s →

p1np2np5np3np1fp2fp5fp3fvs1vs2

vs1vs2

p5n

p2np5f p2f

p1f

p1n

p3n

p2n

Fig. 20: Plot of normal contact and friction impulses when Ais moved to the right, normalized by the normal impulse of A.The subscripts n and f indicate the normal contact impulsesand friction impulses. The sliding velocity between A and Bis vs2 and that between B and the base is vs1.

calculate the forces between different parts in the assembly(Figure 20). When a force high enough to move A to the rightis applied, it begins to slide and drags B along with it dueto imbalance of friction forces between the top and bottomsurfaces of B. Since movement of A induces movement inB, the disassembly sequence is unstable. If we invert thisdisassembly sequence, then the assembly sequence is B-A.When A is inserted horizontally over B from the right, thesliding motion of B is prevented by the left wall, and hence itis stable. Thus, when the horizontal path is selected, assemblyB-A and disassembly A-B are not invertible.

IX. CONCLUSION

The focus of this paper is the influence of part motion andassembly sequence on assembly stability. Given this goal, wego beyond the prior approach of performing a static analysis ofthe disassembly tree. We instead use linear complementarityto perform the stability analysis at each step of motion in thedisassembly process as parts are removed sequentially. Weshowed that disassembly sequences that are consistent withonly geometric contraints may be infeasible in the presenceof gravity or friction. Furthermore, stability analysis can beused to identify and prevent instability inducing motions byintroducing fixtures, by selecting alternative motion paths,

12

(a) (b)

Fig. 18: Plot of normalized force versus time for disassembly of the assembly of Figure 17. All forces are normalized withrespect to the weight of A. (a) B is moved to the left. At 10 s, A becomes unstable, as shown by positive accelerations atcontact points 7, 33 and 38. (b) Subassembly A-B is moved to the left. The gripper grasps and moves A, which pushes B tothe left. The disassembly is stable.

or by alternative disassembly sequences. Previously, onlythe disassembly sequence had been shown to be importantfor stable disassembly. We have shown that not only thedisassembly sequence, but the motion paths of the parts alsocan affect the stability of the disassembly process. We exploredthe interplay of motion path and sequence for both 2D and3D examples. We further proved conditions for motion pathdependence in disassembly problems. We additionally provedthat our LCP based stability analysis and the prior AND/ORgraph based static stability analysis [25] yield the same resultfor path-independent disassemblies. We then extended stabilityanalysis of single part disassembly to stability analysis of sub-assemblies. We also showed that assembly and disassembly,invertible when only geometric constraints are considered, canbecome noninvertible in the presence of friction.

ACKNOWLEDGMENTS

Thanks to Nilanjan Chakraborty, Raju Mattikalli, and JeffTrinkle for informative discussions, and to Jedediyah Williamsfor technical assistance.

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Sourav Rakshit received the Ph.D. degree and M.E.degree from the Department of Mechanical Engi-neering at the Indian Institute of Science, Bangalore,India in 2012 and 2006. He received his B.E. inMechanical Engineering from Jadavpur University in1999. He was a Scientific Officer at Bhabha AtomicResearch Centre (BARC), India from 1999 to 2004,and a Postdoctoral Researcher at UNC Charlottefrom 2012 to 2014. He is an Assistant Professorat the Indian Institute of Technology, Madras, In-dia. His current research interests include multibody

mechanics, biomechanics, assembly planning, robotics, and optimizationalgorithms.

Srinivas Akella (S ’90, M ’00) received his Ph.D.in Robotics from the School of Computer Science atCarnegie Mellon University in 1996, and his B.Tech.from the Indian Institute of Technology, Madras in1989. He is a faculty member in the Department ofComputer Science at the University of North Car-olina at Charlotte. He was previously on the facultyof the Computer Science department at RensselaerPolytechnic Institute, and was a Beckman Fellowat the Beckman Institute, University of Illinois atUrbana-Champaign prior to that. Dr. Akella is a

recipient of the National Talent Search Scholarship from the Governmentof India, and of the NSF CAREER award. His research interests are indeveloping geometric and optimization algorithms for robotics, automation,and biotechnology applications.