Search Strategies for Peg-in-Hole Assemblies with Position Uncertainty

Siddharth R. Chhatpar1 and Michael S. Branicky2

Case Western Reserve University10900 Euclid Ave, Cleveland, OH 44106-7221, U.S.A

{ src2, msb11} @po.cwru.edu

1 Corresponding Author. Ph.D. Student, Mechanical and Aerospace Engineering Dept., Glennan 109.2 Asst. Prof., Electrical Engineering and Computer Science Dept., Glennan 515B.

Abstract. This paper presents search strategies for peg-in-hole assemblies with position uncertainty. The assemblyof parts whose position uncertainty far exceeds assemblyclearance has to rely on either visual assistance or blindsearching to achieve part mating. In the absence of visualassistance we look at blind search strategies to cover thesearch area in an efficient manner. The paper alsodescribes a tilt strategy using special cues to guide theassembly and enhance blind search. Results fromassemblies on an actual robot are presented.

1. IntroductionFor the class of peg-in-hole assemblies, where the relativeposition uncertainty of the assembling parts exceedsassembly clearance, focus shifts from the dynamics of theassembly to the problem of searching for part alignment.Further, if machine vision is not helpful in the search forpart alignment then the assembly has to rely on blindsearch strategies. This problem then lies more under thearea of motion planning.

A significant amount of research has been carriedout in the field of peg-in-hole insertion. The initialconcentration was on pegs of circular cross-section. Themajor contributions of Whitney and Asada assumedchamfered holes. Whitney [1] segregated the assemblyprocess into four parts: chamfer crossing, single-pointcontact, two-point contact and insertion. Quasi-staticconditions were developed for each part of this assemblyprocess. The RCC (remote center compliance) device wasdesigned based on this analysis. It places the center ofcompliance of the peg at its tip, so that any moment aboutthis point causes the peg to correct its orientation relativeto the hole. Asada [2] analyzed the dynamic behavior ofthe peg in high-speed insertion. A special insertion hand(dynamic RCC) was designed on the basis of the dynamicanalysis. Strategies for unchamfered peg-in-hole insertionwere proposed by Inoue [3] and Simunovic [4]. Inouestarts with the peg tilted with respect to the hole, whichinsures an initial configuration with some part of the peginside the hole. The center of the hole is then inferred bymoving in orthogonal directions and recording the hole’sboundaries. Simunovic infers the exact position of thehole, with respect to the peg, from the measured momenton the peg about some axis in the horizontal plane. This

moment arises from small misalignments between peg andhole positions resulting in the peg overhanging the holewhile pressing down on it. Recent work by Caine andStrip has focused on rectangular peg-in-hole insertions.Caine [5] selects an initial configuration with the peg tiltedabout all three axes, and then leads the assembly through aseries of configurations that sequentially reduce thenumber of degrees of freedom of the peg in contact withthe hole and thus reduce the positional uncertainty of thesystem. Strip [7] extends this strategy in combination witha general insertion strategy developed in [6] to generalizeit for all convex three-dimensional pegs. The strategystarts with a tilted peg and the remainder is based roughlyon the approach used by humans in inserting tight-fittingparts. The peg is moved in contact with the hole withtranslational compliance perpendicular to the direction ofthe applied force and rotational compliance about the axisin line with the direction of insertion. But both Caine andStrip assume close initial alignment of peg with hole.According to Strip [7]: “Simple fixturing or a crude visionsystem should allow us to align the z-axis position towithin 0.25 inches and rotational angular displacementwithin a degree or two.”

In this paper we tackle the problem ofovercoming gross misalignment of peg and hole to achieveassembly. Our problem is motivated by actual vehiculartransmission assemblies, where large position uncertaintiescan be expected in spite of precise fixturing of parts, asexplained below. Moreover, the particular construction ofthese assemblies prevents the use of vision to achievenominal alignment.

More recent research has concentrated ondeveloping higher-level, intelligent force-responsivestrategies. McCarragher [8,9] successfully combinedcompliant manipulation with a higher-level planning basedon the discrete event systems theory [10].

The problem of motion-planning underuncertainty has been extensively covered by Latombe. In[11], Latombe tackles the motion planning problem withuncertainty in control and sensing. It relies on the generalapproach of “preimage backchaining,” which wasoriginally proposed in [12].

Newman et al. [13] reported on an impedance-based low-level algorithm and its interface to higher-level

strategies to perform assembly tasks involving largeposition uncertainties.

Chhatpar and Branicky [14] applied a hybridsystems approach to build a general framework that allowsvarious search strategies to be combined at a higher-levelof planning and yet be compatible with low-levelcompliance control.

Our goal in this paper is twofold. First, to find aformal description of the uncertainty and the requirementsfor a candidate search strategy. And secondly, to proposevarious search strategies meeting different requirementsimposed by the need to achieve fast assemblies usingrobots. We use the specific task of assembling ourexample transmission components to focus thisinvestigation.

The paper is organized as follows. Section 2describes the example real-world assemblies and therelated uncertainty issues. Sections 3 and 4 present ouranalysis of the search space and various example searchtrajectories. Section 5 gives implementation details andresults on the simulations and actual robotic assemblies.In Section 6 we discuss future work.

2. MotivationOur over-arching goal is to automate the assembly ofvehicular transmission components such as the insertion ofa spline-toothed hub within a “ forward clutch” assembly(Figure 1). The forward-clutch housing, shown in Figure1(b), consists of a cylinder with five internal rings (clutchplates), each with spline teeth about the inner radius.These rings are each capable of sliding independently inthe horizontal plane by up to 1.6mm and rotating about avertical axis. A cylindrical hub with teeth on its outersurface, shown in Figure 1(a), is to be inserted within thehousing of Figure 1(b). However, the clutch plates are not,in general, aligned either radially or tangentially. Theradial assembly clearance between the hub and a ring is0.23mm. Thus, the radial assembly uncertainty is morethan six times larger than the clearance required forsuccess. Additionally, the hub must align with the z-rotation of each spline ring within 5 mrads for successfulassembly, even though the tooth spacing is 160 mrads.The uncertainty in the angular position of the spline ringsis arbitrarily large (though periodic with the tooth pitch).Using machine vision to measure the translational androtational displacements of the five rings is impractical,since the rings are stacked, occluding a good view. In

manual assembly, the approach used is to exert adownwards force while rotating and translating the hub,searching for the x-y position and z-rotation for successfulassembly.

3. Discretization of Search SpaceWe take the search in the x-y plane to configuration spaceby reducing all the dimensions by the peg diameter (i.e.considering the peg to be a point). See Figure 2.

The reduced radius of the hole becomes theassembly clearance, τ. The search area, H, is the area ofmobility of the center of the hole. From this configuration-space viewpoint, finding x-y coordinates of the hub thatsatisfy adequate concentricity with a ring (to within theassembly clearance) corresponds to finding coordinatesthat lie within a 0.23mm diameter circle located anywherewithin a 1.6mm diameter bound. We can see that the pegpoint need not sweep the entire search area. If a pointcovered by the peg fails to lead to peg-hole alignment, thisensures that the hole center does not lie within τ/2 distanceof that point. Thus, the search area is discretized by theassembly clearance, i.e. for every point the peg visits itcovers a search area equaling the assembly clearance.Strictly speaking, the peg only needs to visit a certain set Aof points, such that no point in the search area is more thanτ/2 distance away from all points in A. This is illustratedin Figure 3.

3.1 Generating Search TrajectoriesSince we are interested in assembling the given parts usinga robot, we apply a restriction that the search trajectory bea continuous path instead of a set of disjoint points. Anyarbitrary continuous path that satisfies the requirement thatno point in the search area is more than τ/2 distance awayFigure 1. Forward clutch assembly

(a) Forward Clutch Hub (b) Housing and spline rings

Search Area1.6 mm (dia)

AssemblyClearance0.23 mm

Peg

HoleCenter

Figure 2. Configuration Space Description

Figure 3. Discretization of the search space

(√2)τ

from all points on the path is a candidate. In other words,a candidate path when dilated by τ/2 units should cover theentire search region. We call this the “spray paint”technique, by analogy to covering an entire surface withpaint while traversing a path. As a counterexample, Figure4 shows a continuous path that is not a true candidate.

Joining Discretized Points: If P is the set of all candidatepaths through the search area, then our problem is to findthe shortest path that contains A. Thus, we have to find theshortest route connecting all the points in A. This isanalogous to the travelling salesman problem (TSP). Thecomputation time for an exact solution to the TSPincreases exponentially with the number of vertices (gridpoints in A), becoming rapidly prohibitive in cost as thevertices increase. To get an approximate solution to theTSP we used simulated annealing. One of the results isshown in Figure 5 below.

Another important restriction we have to place onthe search trajectory is that the path be as smooth aspossible (large radius of curvature). For instance, the pathshown in Figure 5 has many corners and for the robot totraverse that path accurately it would need to decelerate toa complete stop at every corner. This would slow therobot down considerably and even though the path is short,the robot will take a long time in following it. Searchtrajectories that are suitable in this respect are describedbelow.Concentric Circles: If the search space is covered withconcentric circles spaced at (2τ) distance radially, then weare assured that no point in the search space is more than τ

distance from all the points on the circles. This path isillustrated in Figure 6.Spiral Path: The spiral search path is efficient in lengthand yet has smooth curvature throughout. The dimensionsof the spiral can be determined from the search area and

the assembly clearance, by setting pitch, p, equal to theassembly clearance, τ:

Using this and fixing a constant path speed, ds/dt = v, wecan derive

Thus, we obtain dr/dt and dθ/dt from the assemblyclearance, and we can set a maximum limit on r to equalthe radius of the search space. The spiral search path waswidely used in actual robotic assemblies of thetransmission components (for more details see [13]).

Before the start of the search, as above, everypoint in the search space has an equal probability ofcoinciding with the location of the hole-center. As thesearch proceeds, the probability of the points within theclearance area of every point visited, reduces to zero (untilthe hole is found). Thus, the region with non-zeroprobability of containing the hole-center shrinks as theassembly progresses. In the next section, we look at astrategy that produces a much greater shrinkage of thepromising region with each point visited.

4. The Tilt StrategyDuring manual assembly of the forward clutch, it wasobserved that humans tip the hub into the hole to infer thedirection of misalignment. This can be explained asfollows. For a circular peg and hole, if the peg has anoverlap with the hole it can tilt into the hole, as shown inFigure 7. The maximum tilt that the peg can achieve isabout the x-x axis and motion perpendicular to this axis, inthe direction of the tilt, will lead to alignment of the pegwith the hole.

��

r pr

p

θ π πθ� � �

2 2� �

�θ

π

��

v

rp2

2

24

Search Area

Continuous Path

Figure 4. Continuous path not a true candidate

Point notcovered

Figure 5. Possible Search Trajectory

2τ

Search Area

Figure 6. Continuous path of concentric circles

Moreover, if the peg is tilted by a specific amountabout a specific axis, then the amount of partial insertion itachieves in the hole (without full alignment) depends onthe relative positions of the peg and the hole (Figure 8(a)).The plots in Figure 8(b) show the insertion of the peg for arange of horizontal positions, when the hole position isfixed and the peg-tilt is fixed at 30°. (The point to keep inmind is that the peg need not have two-point contact with

the hole. In fact, in most conditions, the peg and holetouch in only one point at intersection.) It can be seenfrom this plot that the mapping from relative position ofpeg and hole to amount of partial insertion is many-to-one,i.e. more than one relative position will yield the samepartial insertion. Hence, the inverse mapping frominsertions to positions is one-to-many. Thus, from oneinsertion measurement we get a set of possible relativepositions with the correct solution contained therein(Figure 9(a)). If we now move the peg by a knowninterval in the horizontal plane (and move vertically asrequired to achieve contact with hole) and take anotherinsertion measurement, it yields another set of possiblepoints. At this time, only the points belonging to both setsare actual candidates. This is illustrated in Figure 9.Figures 9(a) and (b) show the solution sets generated fromrespective insertion measurements. The intersection of the

solution sets (Figure 9(c)) yields two points representingthe possible location of the hole. A third insertionmeasurement yields yet another solution set (Figure 9(d))and, of the two solutions carried over from the previousintersection, the one that belongs to this third set, is theactual hole-location. Thus, generally, on successiveinsertion measurements, the set of candidate points keepsshrinking.

While the circular peg-in-hole is a 2-dimensionalsearch problem, the various transmission componentassemblies that we are interested in, generally requiresearches in higher dimensions. Our goal is to apply thistilt strategy to infer alignment coordinates for a generalassembly of the type of our example transmissioncomponents.

To advance the strategy to higher dimensions, wenext considered the problem of assembling a square peg ina square hole. This problem requires not only planarposition alignment of peg and hole, but also orientation(rotation about the vertical axis) alignment withinassembly clearance. Thus, it poses a search in 3-dimensions: relative position in the plane and relativeorientation about the vertical axis. Adopting a similarapproach as with the circular peg-in-hole, we consider thepartial insertion of a tilted square peg in a square hole.The peg is tilted about an axis coinciding with one of thediagonals as shown in Figure 10. The partial insertions ofthe tilted peg are calculated for various relative horizontal(x, y) positions of the peg and hole and various relativeorientations (� ). Particular examples are plotted in Figure11. The top figure is for the peg and hole aligned in

Hole

Peg

No clues

HolePeg

Small tilt @ x-x

x

HolePeg

Large tilt @ x-x

x x

x

Figure 7. Infer position of hole from tilt

Figure 8(a). Tilted peg partial insertion in hole

Figure 8(b). Plot of partial insertions for allrelative (x,y) positions

Peg

Hole

Top view Bottom view

Hole

Peg

Figure 9. Inverse mapping: Insertion → Positions

Intersection ofsolution sets

(c) (x,y) solutions

Insertion 3: 0.60mm

(d) actual hole-center

Insertion 1: 1.24mm

peg solutions hole(a)

Insertion 2: 0.38mm

peg new position

(b)

orientation; in the bottom figure, they have a relativeorientation of π/4.

The inverse mapping that we desire from theseplots is again one-to-many, i.e. for each given partialinsertion there are many possible relative (x, y, � )combinations. Hence, a single insertion measurement willnot yield the precise (x, y)-location and -orientation of thehole required for assembly. Instead, we get a set of these(x, y, )’s, each a possible solution point in 3-dimensionalspace. Example solution sets from an actual simulation areshown in Figure 12. Our strategy is to then move the pegby a known interval in (x, y) and take another insertionmeasurement, which yields another set of points. At thattime, only the solution points belonging to both sets areactual candidates. Thus, an intersection of the sets yields asmaller set of possible points. We may continue takinginsertion measurements at various positions to finallyarrive at a single (x, y, � ) combination.

Now, to apply the tilt strategy to any arbitrarypeg-in-hole assembly of the type of the exampletransmission components, we need to establish a mappingfrom the misalignment (search) space to partial insertionsof the peg in the hole. Unlike the above examples, withcircular and square cross-sections, we assume that thismapping will not be computed offline (with knowledge ofpart geometry), but will have to be established (learnt) byprior experimentation with the particular assembly. Thelearning can be automated with the help of a robot, so thateach point in the misalignment space is visited and thepartial insertion noted. The mapping can be set up in theform of a discretized table as was done with the exampleassemblies above. Once the full mapping has beenestablished, the inverse map would involve a simple tablelookup. From there, the tilt strategy could be applied toperform the actual assembly, so that successive insertionmeasurements lead to the determination of themisalignment.

5. ResultsThe blind strategy of moving the peg in a spiral trajectoryto search for alignment with hole (described in Section3.2) has been extensively used for the assembly of manytransmission components (see [13] for details).

Simulations were set up for testing the tiltstrategy on both the circular and square peg-in-holes. Themapping from search space (circular:2-dimensional,square:3-dimensional) to partial insertions was generatedand stored in the form of a table. The inverse mappingwas then used in the application of the tilt strategy todetermine hole location. In generating the map, thecontinuous search space was approximated withdiscretized points. Due to this discretization, thecomputation of set intersections was carried out using aEuclidean distance metric to compare points. If two pointslie within a set (tolerance) distance of each other they areconsidered the same. The simulations were successful forboth the circular and square cases.

In attempting the circular peg-in-hole assemblyon a robot, the following departure from the tilt strategy

z

z

x

x

d a

c b

hPEG

HOLE

Figure 10. Peg tilted about its diagonal axis

Figure 11. Peg Elevation for various relative(x-y) positions

Peg-Hole alignedin orientation

Peg-Holeangle = π/4

Figure 12. Successive solution sets generated fromindividual insertion measurements (same hole setup)

- Each rectanglerepresents variousrelative (x-y)positions and oneorientation (� ).

- Solution set isdiscretized in .

- Circled pointrepresents truecombination

was employed. Instead of making individual insertionmeasurements, the maximum-tilt axis x-x (refer Fig. 6) wasdetermined through peg precession, which gives thedirection of motion for alignment. Also, the maximuminsertion value was used to determine the amount ofmisalignment along this direction. The experiment isdescribed below.

A hole of 100mm diameter and a peg of 99.5mmdiameter were used for the assembly tests. Thus, theproblem is finding alignment coordinates that lie within a0.5mm-diameter circle located somewhere within a100mm-diameter bound. A blind spiral strategy wouldrequire a trajectory of length more than 10m to cover theentire region. To begin the experiment the peg and holeare given some (unknown) initial overlap. The peg is tiltedabout some horizontal axis by a small angle (15° was usedfor the experiments), and is made to contact the holesurface. Now the peg is taken through a 360° precession.As the contact point passes over the hole, the peg willlower into the hole. A record of the peg height ismaintained. The tilt axis corresponding to the lowestheight achieved by the peg is noted; this is the x-x axisshown in Figure 6. Motion perpendicular to this axis inthe direction of the tilt leads to peg-hole alignment.

In over 60 trials of the assembly, we had 100%success. The average time for the assemblies was 7.1seconds, compared to a worst-case time for a spiral searchof 40s. For details see [15].

To test the tilt strategy on a square peg-in-hole,we used a 70mm edge square hole with a 69.5mm edgesquare peg. The insertion-to-(x, y, � ) mapping generatedfor the simulation was used with successive insertionmeasurements to find the correct position and orientationof the hole. But the discretization of the (x, y, � ) spaceresulted in substantial errors, especially in the orientation,and when combined with the general errors introduced byan actual robotic assembly, the robustness of the strategywas reduced considerably. Currently, we use this strategyto acquire a good initial approximation to the hole-locationand then shift control to a more traditional strategy toachieve assembly. The traditional strategy uses acompliant mode to accommodate the remaining errors.Further experimentation and reliability tests are pendingfor the assembly.

6. Conclusions and Future WorkThis paper has presented a case study in search strategiesfor peg-in-hole assemblies with position uncertainty. Wegave a configuration space description of the search space,its peculiar discretization by the assembly clearance, andthe spray paint coverage of the search area. The paperdescribed various blind search strategies to cover thesearch area in a time-efficient manner with particularrestrictions brought in by practical demands of roboticassembly. The paper also described the tilt strategy toguide the assembly.

In future work, we intend to generalize theapplication of the tilt strategy to our example transmissioncomponent assemblies. Another idea that we intend tofocus on is finding intelligent next measurement locations.That is, in making successive measurements to shrink thesolution set, is there an intelligent way to make the nextmeasurement that would lead to maximum benefit in termsof reduction of the elements in the solution set and fasterconvergence on the correct solution?

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4. S. Simunovic, An Information Approach to PartsMating, Sc.D. Thesis, MIT Mech. Eng. Dept, 1979.

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