Targeted Sampling Methods: A Literature Review

Targeted Sampling Methods: A Literature Review

John Baxter

December 4, 2012

Definition. Let Cseed be a configuration in configuration space C. Let D(Cx, Cy) be
a distance metric over element pairs in C. Targeted Sampling seeks to efficiently
find Cvariant ∈ C such that D(Cseed, Cvariant) satisfies specified validity constraints.

Introduction

Targeted sampling problems appear extensively as subproblems in larger motion
planning problems. One common subclass of targeted sampling problems (hereafter
referred to as Bounded Sampling) requires that new configurations are found
between upper and lower distance bounds of an original configuration. Bounded
sampling problems are critical subcomponents of complete and resolution-complete
deterministic motion planning algorithms, and also appear frequently in heuristic
and probabilistic motion planning algorithms.

Resolution-complete sampling is most commonly performed in RN, as bounded
sampling problems are particularly easy to solve exactly in this space. In general,
bounded sampling is not trivial; significant effort is required to prove resolution-
complete coverage of SE(3), the space of rigid transformations [1], for example.

Targeted sampling is a special case of more general Inverse Configuration
Space problems, which attempt to find subsets of a configuration space satisfying
arbitrary constraints. An approach for approximating solutions to − δ continuous
inverse configuration space problems for rigid bodies in terms of simpler bounded
sampling problems is presented in [1].

1

Motivation

Many common robotic design patterns, including manipulator arms and snake-
like robots, can be modeled as serial kinematic chains; polymers and protein may
also be efficiently modeled as kinematic chains. Kinematic chains are commonly
represented within computations as a recursive hierarchy of matrix transformations,
or equivalently by the individual translations and rotations transforming points in
a parent link’s object space to that of its child. Motion planning algorithms for
kinematic chains typically choose the non-constant translations and rotations as
configuration space dimensions.

The non-linear influence of kinematic degrees of freedom on overall structure
makes certain motion-planning tasks more difficult. Resolution-complete sampling
approaches based around ensuring resolution-completeness in the configuration space
dimensions are not satisfactory for use with kinematic chains, as each degree of
freedom’s structural influence varies significantly as a function the configuration-
dependent positions of the associated joints.

The Root Mean Square Distance (RMSD) is a common metric describing
the structural similarity between pairs of objects in terms of points of correlation on
kinematic chains. In computational biology, structural alignment between protein
conformations is often performed by finding rigid body transformations minimizing
RMSD. Additionally, several well-studied techniques for observing the average RMSD
of real-world molecular ensembles exist, including X-Ray Diffraction (XRD) and
Nuclear Magnetic Resonance (NMR). RMSD therefore provides an important
measure for assessing the physical verisimilitude of molecular simulations.

Sampling approaches capable of generating variant kinematic structures with
RMSD from an original exemplar within user-provided bounds would be more easily
compared to the R3 coordinates typical of advanced, computationally expensive all-
atom force-field approaches to molecular dynamics.

Statistical Geometry

The geometric and statistical properties of kinematic chains have been extensively
explored in simple mathematical models. The simplest do not impose constraints
on self-collision; nevertheless, they have been effective in describing the stochastic
behaviors of many common polymer and protein phases [2].

2

In the freely-jointed chain model, a chain composed of equal length linear
segments are connected end-to-end serially, with no constraints placed on the relative
orientation of one segment to the next. It is relatively straightforward to approximate
the probability distribution of several characteristic lengths for random freely-jointed
chains; for example, the average distance between the end points is proportional to
the square root of the number of links [2]. The continuous, infinitesimal segment
length limit of the freely-jointed chain model is the ideal chain model.

The freely-rotating chain model fixes the angle between one kinematic segment
and the next to a constant; rotation is allowed only about the axis of each linear
segment. The end-to-end distance in this case scales not only with the square root
of the link count, but also with a trigonometric function of the angle between each
link. The continuous analog is known as the worm-like chain model.

Statistical geometric approaches have been applied only sparsely to probabilistic
motion planning. One example is [3], in which normal mode analysis (NMA) is
used to drive a molecular model where the degrees of freedom are the low-frequency
normal modes. Because NMA approximation accuracy degrades with the structural
distance from the conformation used in the NMA calculations, [3] is forced to perform
many NMA calculations in order to sample large portions of configuration space.

Implicit Evaluation

Conceptually, the simplest possible approach to targeted sampling is to directly
search the configuration space. Because the search spaces can be astronomically
large, such an approach may seem completely infeasible. If the distance metric is well-
behaved over the configuration space, however, iterative optimization approaches
may be completely sufficient. However, to date there does not seem to have been
any general analysis of how the parameterization of different restricted subclasses of
kinematic structures influence the behavior of such implicit functions.

If the distance metric function is not amenable to iterative optimization, it may
still be feasible to solve bounded sampling problems using a general-purpose root-
finding algorithm. Such algorithms are well-studied as a method for efficiently ray-
tracing implicit functions. A computationally intensive but very general approach
that covers ray-tracing arbitrary-form implicit surfaces is presented in [4]; it may
be that more efficient specialized ray-tracing algorithms could be applied to specific
combinations of distance metric and configuration parameterization.

3

General Biased Sampling

An alternative approach to implicit function evaluation would be to use existing
configuration space sampling methods, by defining an alternative obstacle definition
based on metric distances between configurations. Several probabilistic sampling
methods have been devised with the intent of placing samples as close to configuration
space obstacles.

Local planner approaches like OBPRM [5] and variants thereof [6] would be in
effect equivalent to linear or binary ray-tracing solutions. The primary advantage of
such methods over ray-tracing would be in terms of implementation simplicity in the
context of a preexisting motion planning library.

More global biased search approaches, such as Gaussian sampling [7], Bridge
Test sampling [8] and biased sampler composition [9] might be applied. However,
these may have trouble finding configurations if a small distance from an original
configuration is required or the bounds on distances are tight relative to the overall
size of the configuration space.

Obstacle-hugging roadmap expansion methods such as the Retraction-based RRT
[10] might be used to iteratively find configurations satisfying narrow distance metric
bounds. For RMSD bounded sampling, such approaches should be capable of finding
alternative solutions to an initial solution that are close in configuration space to
the initial solution, which could be very useful for searching for improvements to
configurations only slightly in collision. However, finding the initial solution would
still require local or global search as described above.

Inverse Kinematics

RMSD bounded sampling has similarities to the problem of randomly sampling
closed kinematic chains. One approach to sampling closed kinematic chains (as in
[11], [12]) is to divide the chain into active and passive portions, and then use inverse
kinematics to constrain points on the active chain to have zero distance from the
paired points on the passive chain. In comparison, in RMSD bounded sampling our
original kinematic structure acts as a passive chain, our variant kinematic structure
is the active chain, and each pair of correlation points is constrained such that the
root mean square distance between them is within user-specified bounds. Closed
chain sampling approaches capable of handling large numbers of kinematic cycles
exist, such as [13], making this method theoretically feasible.

4

More light-weight bounded sampling for structural metrics might be developed
from continuum-robot inverse kinematics. A motion planning algorithm based on
inverse kinematics for steerable robotic needles [14] presents an elegant approach to
motion planning with circular arc path segments of controllable curvature. They
further present a null-motions (see [14], Figure 9) approach to generating variant
needle paths with identical end-point positions and orientations. This is particularlly
significant because of the geometric similarities between the structure of their paths
and worm-like chains.

There should be generalizations of null-motions producing variant configurations
with non-zero changes to end-point positions and orientations; a recursive application
of such motions should be able to implement RMSD bounded sampling cleanly and
accurately, at least in principle. The Paden-Kahan subproblems approach to inverse
kinematics (reviewed in [15], chapter 3 section 2) might be a fruitful theory upon
which such a recursive approach might be developed. The following are examples of
common subproblems:

Subproblem 1. (Rotation About One Axis). Given points p1 and p2 and axis
ξ, solve for angle θ in range [0, 2π) such that point p1 rotated about axis ξ by angle
θ is point p2. (At most 2 solutions.)

Subproblem 2. (Rotation About Two Subsequent Axes). Given points p1
and p2 and axes ξ1 and ξ2, solve for angles θ1 and θ2 in range [0, 2π) such that p1
rotated about ξ1 by θ1 and then rotated about ξ2 by θ2 is p2. (At most 4 solutions.)

Note: In the simpler case, it is required that ξ1 and ξ2 also intersect; however, [16]
presents an extension to arbitrary non-intersecting axes.

Subproblem 3. (Rotation to a Specific Distance). Given points p1 and p2,
axis ξ, and distance δ, solve for angle θ in range [0, 2π) such that p1 rotated about ξ
by θ is exactly δ from p2. (At most 2 solutions.)

5

Inverse Kinematics Algorithm

An algorithm to exhaustively find all possible solutions to the bounded RMSD
sampling problem as in terms of the Paden-Kahan subproblems is unlikely to be
tractable; even the inverse kinematics of positioning a single end effector requires
special robot design restrictions for tractable analytical solutions above about six
degrees of freedom. If regular inverse kinematics problems positioning an end effector
are thought of as optimization problems, the exact RMSD sampling problem is the
multiple objective optimization version of that problem.

Still, the subproblems might be used to find a fraction of the total number of
solutions. One possible algorithm for finding a subset of the solutions to the bounded
RMSD sampling problem (without RMSD alignment) for a fixed-base kinematic
chain with N correlation points might involve repeatedly applying Subproblem 3.
To simplify the presentation, the algorithm outline presented below instead finds
a subset of the solutions to the bounded Mean Distance sampling problem; an
RMSD variant should be possible by choosing proper scaling values for the distance
targets of each subproblem.

Suppose a mean distance target δ1. Start by moving the point of correlation

closest to the fixed base a distance of δ1, if possible. If the correlation point can not be

moved a distance δ1, one can take a variety of actions, such as choosing the maximum

distance possible. Call the difference between δ1 and the point of correlation’s offset

from its old position r1. Now recurse on the next point of correlation with the axes
r2
beyond that point of correlation and a mean distance target δ2 = δ1 + N −1 . Note

that in recursion, correlation points are moved to within distances from their original

positions, not their positions at the beginning of that step of recursion. Also note

that in recursion step k, the mean distance target is δk+1 = δk + rk , except of
N −k
course for the terminal step where N = k.

In the above algorithm, if every step of the recursion meets its target, the resulting
configuration will have a mean distance of exactly δ from the original configuration.
Furthermore, small failures in the early stages of the recursion are capable of being
fixed by later steps of the recursion. However, the completeness of the algorithm is
to some degree limited by its greediness; it is possible that the heuristic used to select
distance subtargets early on will make certain valid solutions requiring completely-
satisfiable correlation points be only partially satisfied impossible. Randomizing the
order of correlation points manipulated is not a solution, as previously optimized
points will be moved out of position by subsequent rotations.

6

Statistical Geometry Algorithm

An alternative approach (that the author of this literature review is in the process
of investigating) focuses on optimizing the influence of each degree of freedom on
all points of correlation, rather than attempting to optimize the position of each
point of correlation in sequence. Given an initial seed configuration, the structural
influence of each degree of freedom on kinematic points of correlation is measured
by modifying the value of that degree of freedom while holding all other degrees of
freedom constant.

The initial approach assumes a kinematic chain or tree composed of rotational
degrees of freedom only. For calculating joint structure influence, rotations of π
are a reasonable choice, as this produces the maximum possible distances (without
alignment) and allows many further optimizations. The RMSD for each joint acting
alone is then calculated and stored. A procedure very much like Subproblem 3 exists
for finding angles in [0, 2π) relative to these stored values (details omitted).

When applied to freely-rotating chain models of uniform inter-segment curvature,
attempting to create a specific fraction (details omitted) of the total desired RMSD
for each degree of freedom produces a mean total RMSD very close to the desired
RMSD, although there is significant variance. Interestingly, the above-mentioned
scaling factor is a simple non-linear function of the number manipulated degrees of
freedom. The variety of possible configurations produced can be increased without
introducing significant error by making the degree of freedom subtargets distributions
with means about their heuristically fraction of the total desired RMSD.

One downside of this approach is that it does not gracefully handle cases where
degrees of freedom can not produce a specified subtarget RMSD. However, it might
be possible to handle such cases by estimating the shortfall that needs to be made
up by other degrees of freedom, in a manner similar to what was presented in the
Inverse Kinematics Algorithm.

Applicability and Synergies

Dimensionality Reduction

Bounded sampling algorithms handling a variable active degrees of freedom could
work well in conjunction with dimensionality reduction sampling approaches based
on conditionally deactivating a subset of a robotic system’s degrees of freedom.

7

The dimensionality reduction approach taken by the Manhattan-like RRT [17]
partitions the degrees of freedom of a system into active and passive degrees of
freedom. Motion planning is initially attempted using only the active degrees of
freedom of a system. Passive degrees of freedom are only manipulated when collision
detection indicates that physical components of the simulated system controlled by
the passive degrees of freedom are impeding the progress of the components controlled
by active degrees of freedom. This could be adapted to use RMSD bounded sampling
with a fixed structural change target per step. Doing so would be helpful in the case
of kinematic chain simulations. Na¨ıve configuration space distance steps do not
produce uniform structural exploration for kinematic chains in general, a problem
amplified if the number of degrees of freedom vary; structural bounded sampling is
a possible solution to this problem.

A force field planning method for highly articulated linkages is presented in [18],
in which joints producing motion below a certain threshold are not moved during
physics time steps. A light-weight variant using bounded sampling in place of force-
based simulation may result in improved runtimes.

Partial Validity Sampling

Structural bounded sampling may also be useful in sampling schemes that move
beyond the standard Boolean validity tests of traditional roadmap approaches. One
approach might be based on the iterative relaxation of constraints approach [19],
which misleadingly actually involves initially starting with fewer constraints than
the true problem and iteratively strengthening constraints once solutions are found
for less constrained variant problems. For kinematic chains, we might initially start
with a solution that only require one segment to not be in collision, and iteratively
increase the required number of segments not in collision until a collision-free path is
found. Structural bounded sampling combined with estimates of penetration depths
of the segments in collision could be used to improve partially valid samples.

Biological adaptive immune systems use clonal selection to improve partial matches
between antigens and complementary structures; immune coverage can be modeled
in terms of reaction distances in shape spaces [20]. A generalization of the Gaussian
[7] and bridge test [8] PRM sampling approaches would be to space uniformly, and
then produce structural variants of partially-valid samples, biased probabilistically
towards almost-valid samples. Such an approach would likely work particularly well
at resolving minor self-collisions in long kinematic chains, but would probably also
be effective in placing chain samples along obstacle boundaries.

8

References

[1] S. Nelaturi and V. Shapiro. Solving inverse configuration space problems by
adaptive sampling. Computer-Aided Design, 45(2):373–382, 2013.

[2] Hiromi Yamakawa. Modern Theory of Polymer Solutions. Harper Row, New
York, New York, 1971.

[3] S. Kirillova, J. Cort´es, A. Stefaniu, and T. Sim´eon. An NMA-guided approach
for computing large-amplitude conformational changes in proteins. Proteins:
Structure, Function and Bioinformatics, 70(1):131–143, 2007.

[4] A. Knoll, Y. Hijazi, A. Kensler, M. Schott, C. Hansen, and H. Hagen. Fast
ray tracing of arbitrary implicit surfaces with interval and affine arithmetic.
Computer Graphics Forum, 28:26–40, 2008.

[5] Nancy M. Amato, O. Burchan Bayazit, Lucia K. Dale, Christopher Jones, and
Daniel Vallejo. OBPRM: An obstacle-based prm for 3d workspaces. Interna-
tional Workshop on Algorithmic Foundations of Robotics (WAFR), pages 155–
168, 1998.

[6] Cindy (Hsin-Yi) Yeh, Shawna Thomas, David Eppstein, and Nancy M. Amato.
UOBPRM: A uniformly distributed obstacle-based prm. IEEE International
Conference on Intelligent Robotics and Systems (IROS), 2012.

[7] Val´erie Boor, Mark H. Overmars, and A. Frank van der Stappen. The Gaus-
sian sampling strategy for probabilistic roadmap planners. IEEE International
Conference on Robotics and Automation (ICRA), 2:1018–1023, 1999.

[8] D. Hsu, T. Jiang, J. Reif, and Z. Sun. Bridge test for sampling narrow pas-
sages with probabilistic roadmap planners. IEEE International Conference on
Robotics and Automation (ICRA), pages 4420–4426, 2003.

[9] Shawna Thomas, Marco Morales, Xinyu Tang, and Nancy M. Amato. Biased
samplers to improve motion planning performance. IEEE International Confer-
ence on Robotics and Automation (ICRA), pages 1625–1630, 2007.

[10] L. Zhang and D. Manocha. An efficient retraction-based RRT planner. IEEE
International Conference on Robotics and Automation (ICRA), pages 3743–
3750, 2008.

9

[11] L. Han and N. Amato. A kinematics-based probabilistic roadmap method for
closed kinematic chains. International Workshop on the Algorithmic Founda-
tions of Robotics (WAFR), 2000.

[12] J. Cort´es, T. Sim´eon, and J. P. Laumond. A random loop generator for planning
the motions of closed kinematic chains using PRM methods. IEEE International
Conference on Robotics and Automation (ICRA), 2:2141–2146, 2002.

[13] Xinyu Tang, S. Thomas, P. Coleman, and N. M. Amato. Reachable distance
space: Efficent sampling-based planning for spatially constrained systems. In-
ternational Journal of Robotics Research, 29(7):916–934, 2010.

[14] V. Duindam, J. Xu, R. Alterovitz, S. Sastry, and K. Goldberg. Three-
dimensional motion planning algorithms for steerable needles using inverse kine-
matics. International Journal of Robotics Research, 29:789–800, 2010.

[15] Richard M. Murray, Zexiang Li, and S. Shankar Sastry. A Mathematical Intro-
duction to Robotic Manipulation. CRC Press, 1994.

[16] Tan Yue-sheng and Xiao Ai-ping. Extension of the second Paden-Kahan sub-
problem and it’s application in the inverse kinematics of a manipulator. IEEE
Conference on Robotics, Automation and Mechatronics, pages 379–381, 2008.

[17] J. Cort´es, L. Jaillet, and T. Sim´eon. Dissassembly path planning for complex
articulated objects. IEEE Transactions on Robotics, 24(2):475–481, 2008.

[18] Russell Gayle, Stephane Redon, Avneesh Sud, Ming C. Lin, and Dinesh
Manocha. Efficient motion planning of highly articulated chains using physics-
based sampling. IEEE International Conference on Robotics and Automation
(ICRA), pages 3319–3326, 2007.

[19] O. Burchan Bayazit, Dawen Xie, and Nancy M. Amato. Iterative relaxation
of constraints: A framework for improved automated motion planning. IEEE
International Conference on Intelligent Robots and Systems, pages 3433–3440,
2005.

[20] R. Hightower, S. Forrest, and A. S. Perelson. The evolution of emergent organi-
zation in immune system gene libraries. Proceedings of the Sixth International
Conference on Genetic Algorithms, 1995.

10