Sensing Intelligence Motion - How Robots & Humans Move - Vladimir J. Lumelsky Part 11 docx

For example, if linkl1in Figure 6.2 happens to be sliding along an obstacle which is not so in this example, it cannot applyany force onto the obstacle, the contact would not preclude th

We will first analyze the PPP arm (Section 6.2), an arm with three ing (prismatic) joints (it is often called the Cartesian arm), and will develop a

slid-sensor-based motion planning strategy for it Similar to the 2D Cartesian arm,the SIM algorithm for a 3D Cartesian arm turns out to be the easiest to visu-alize and to design After mastering in this case the issues of 3D algorithmicmachinery, in Section 6.3 we will turn our attention to the general case of an

XXP linkage Similar to the material in Section 5.8, some theory developed in

Section 6.2.4 and Sections 6.3 to 6.3.6 is somewhat more complex than most ofother sections

As before, we assume that the arm manipulator has enough sensing to sensenearby obstacles at any point of its body A good model of such sensing mech-anism is a sensitive skin that covers the whole arm body, similar to the skin onthe human body Any other sensing mechanism will do as long as it guaranteesnot missing potential obstacles Similar to the algorithm development for the 2Dcase, we will assume tactile sensing: As was shown in prior chapters, the algo-rithmic clarity that this assumption brings is helpful in the algorithm design Wehave seen in Sections 3.6 and 5.2.5 that extending motion planning algorithms

to more information-rich sensing is usually relatively straightforward Regardingthe issues of practical realization of such sensing, see Chapter 8

6.2 THE CASE OF THE PPP (CARTESIAN) ARM

The model, definitions, and terminology that we will need are introduced inSection 6.2.1 The general idea of the motion planning approach is tackled inSection 6.2.2 Relevant analysis appears in Sections 6.2.3 and 6.2.4 We formu-late, in particular, an important necessary and sufficient condition that ties thequestion of existence of paths in the 3D space of this arm to existence of paths

in the projection 2D space (Theorem 6.2.1) This condition helps to lay a dation for “growing” 3D path planning algorithms from their 2D counterparts.The corresponding existential connection between 3D and 2D algorithms is for-mulated in Theorem 6.2.2 The resulting path planning algorithm is formulated

foun-in Section 6.2.5, and examples of its performance appear foun-in Section 6.2.6

6.2.1 Model, Definitions, and Terminology

For the sake of completeness, some of the material in this section may repeat thematerial from other chapters

Robot Arm. The robot arm is an open kinematic chain consisting of three links,

l1,l2, andl3, and three joints, J1,J2, andJ3, of prismatic (sliding) type [8] Joint

axes are mutually perpendicular (Figure 6.2) For convenience, the arm endpoint

P coincides with the upper end of link l3 PointJ i,i = 1, 2, 3, also denotes the

center point of jointJ i, defined as the intersection point between the axes of link

l i and its predecessor JointJ1 is attached to the robot baseO and is the origin

Figure 6.2 The work space of a 3D Cartesian arm:l1, l2, and l3 are links;J1, J2, and

J3 are prismatic joints;P is the arm endpoint Each link has the front and rear end; for

example,J3 is the front end of linkl2 O1 ,O2 , andO3 are three physical obstacles Also shown in the plane (l1, l2) are obstacles’ projections The cube abcodefg indicates the

volume whose any point can be reached by the arm endpoint.

of the fixed reference system Valuel i also denotes the joint variable for link l i;

it changes in the rangel i = [l i min , l i max] Assume for simplicity zero minimumvalues for alll i,l i = [0, l i max]; alll i max are in general different

Each link presents a generalized cylinder (briefly, a cylinder )—that is, a rigid

body characterized by a straight-line axis coinciding with the corresponding jointaxis, such that the link’s cross section in the plane perpendicular to the axis doesnot change along the axis A cross section of link l i presents a simple closedcurve; it may be, for example, a circle (then, the link is a common cylinder), arectangle (as in Figure 6.2), an oval, or even a nonconvex curve The link crosssection may differ from link to link.2

The front ends of links l1andl2coincide with jointsJ2 andJ3, respectively;the front end of link l3 coincides with the arm endpoint P (Figure 6.2) The

opposite end of linkl i , i = 1, 2, 3, is its rear end Similarly, the front (rear) part

of linkl i is the part of variable length between jointJ i and the front (rear) end

of the link When jointJ i is in contact with an obstacle, the contact is considered

to be with linkl i−1

2 More precisely, we will see that only linkl3 has to be a generalized cylinder to satisfy the motion planning algorithm; linksl andl can be of arbitrary shape.

For the sensing mechanism, we assume that the robot arm is equipped with

a kind of “sensitive skin” that covers the surfaces of arm links and allows anypoint of the arm surface to detect a contact with an approaching obstacle Othersensing mechanisms are equally acceptable as long as they provide informationabout potential obstacles at every point of the robot body Depending on thenature of the sensor system, the contact can be either physical—as is the casewith tactile sensors—or proximal As said above, solely for presentation purposes

we assume that the arm sensory system is based on tactile sensing.3

The Task. Given the start and target positions, S and T , with coordinates

S = (l1S , l2S , l3S ) and T = (l1T , l2T , l3T ), respectively, the robot is required

to generate a continuous collision-free path from S to T if one exists This may

require the arm to maneuver around obstacles The act of maneuvering around

an obstacle refers to a motion during which the arm is in constant contact with

the obstacle Position T may or may not be reachable from S; in the latter case

the arm is expected to make this conclusion in finite time We assume that thearm knows its own position in space and those of positionsS and T at all times.

Environment and Obstacles. The 3D volume in which the arm operates is the

robot environment The environment may include a finite number of obstacles.

Obstacle positions are fixed Each obstacle is a 3D rigid body whose volume andouter surface are finite, such that any straight line may have only a finite number

of intersections with obstacles in the workspace Otherwise obstacles can be ofarbitrary shape At any position of the arm, at least some motion is possible Toavoid degeneracies, the special case where a link can barely squeeze betweentwo obstacles is treated as follows: We assume that the clearance between theobstacles is either too small for the link to squeeze in between, or wide enough

so that the link can cling to one obstacle, thus forming a clearance with theother obstacle The number, locations, and geometry of obstacles in the robotenvironment are not known

W-Space and W-Obstacles. The robot workspace (W-space or W) presents

a subset of Cartesian space in which the robot arm operates It includes the

effective workspace, any point of which can be reached by the arm end effector

(Figure 6.3a), and the outside volumes in which the rear ends of the links mayalso encounter obstacles and hence also need to be protected by the planningalgorithm (Figure 6.3b) Therefore, W is the volume occupied by the robot arm

when its joints take all possible values l = (l1, l2, l3), l i = [0, l i max],i = 1, 2, 3.

Denote the following:

• v i is the set of points reachable by point J i,i = 1, 2, 3;

• V i is the set of points (the volume) reachable by any point of linkl i Hence,

3 On adaptation of “tactile” motion planning algorithms to more complex sensing, see Sections 3.6 and 5.2.5.

Figure 6.3 (a) The effective workspace of the 3D Cartesian arm— the volume that can

be reached by the arm endpoint — is limited by the cubicle abcodefg (b) Since the rear

end of every link may also encounter obstacles, the workspace that has to be protected

by the planning algorithm is larger than the effective workspace, as shown.

• v1 is a single point,O;

• v2 is a unit line segment, Oa;

• v3 is a unit square, Oabc;

• V1is a cylinder whose (link) cross section iss1and whose length is 2l1 max;

• V2is a slab of length 2l2 maxformed by all possible motions of the front andrear ends of linkl2 within the joint limits ofl1and l2;

• V3 is a “cubicle” of height 2l3 max formed by all possible motions of thefront and rear ends of linkl3 within the joint limits ofl1,l2, andl3.The total volume V W of W -space is hence V W = V1∪ V2∪ V3 Out of this,the set {l} = {l ∈ [0, lmax]}, where lmax= (l1 max, l2 max, l3 max), represents points

reachable by the arm end effector;{l} is a closed set.

An obstacle in W -space, called W-obstacle, presents a set of points, none of

which can be reached by any point of the robot body This may include someareas of W -space which are actually free of obstacles but still not reachable

by the arm because of interference with obstacles Such areas are called the

shadows of the corresponding obstacles A W -obstacle is thus the sum of volumes

of the corresponding physical obstacle and the shadows it produces The word

“interference” refers here only to the cases where the arm can apply a force tothe obstacle at the point of contact For example, if linkl1in Figure 6.2 happens

to be sliding along an obstacle (which is not so in this example), it cannot applyany force onto the obstacle, the contact would not preclude the link from theintended motion, and so it would not constitute an interference.W -obstacles that

correspond to the three physical obstacles—O1,O2, andO3—of Figure 6.2 areshown in Figure 6.4

C-Space, C-Point, and C-Obstacle. The vector of joint variables l =

(l1, l2, l3) forms the robot configuration space (C-space or C) In C-space,

the arm is presented as a single point, called the C-point The C-space of our

Cartesian arm presents a parallelepiped, or generalized cubicle, and the mapping

W → C is unique.4For the example of Figure 6.2, the correspondingC-space is

shown in Figure 6.5 For brevity, we will refer to the sides of theC-space cubicle

as its floor (in Figure 6.2 this is the side Oabc), its ceiling (side edgf), and its walls, the remaining four sides C-obstacle is the mapping of a W -obstacle into

C In the algorithm, the planning decisions will be based solely on the fact of

con-tact between the links and obstacles and will never require explicit computation

of positions or geometry ofW -obstacles or C-obstacles.

M-Line, M-Plane, and V-Plane. As before, a desired path, called the main line

(M-line), is introduced as a simple curve connecting points S and T (start and

target) inW -space The M-line presents the path that the arm end effector would

4 In general, the mappingW → C is not unique In some types of kinematics, such as arm manipulators

with revolute joints, a point inW may correspond to one, two, or even an infinite number of points

inC [107].

Figure 6.4 TheW -obstacles produced by obstacles shown in Figure 6.2 consist of the

parts of physical obstacles that intersectW -space plus their corresponding shadows.

follow if no obstacles interfered with the arm motion Without loss of generality,

we assume here that the M-line is a straight-line segment We will also need two

planes, M-plane and V-plane, that will be used in the motion planning algorithm

when maneuvering around obstacles (see Figures 6.7 and 6.9):

• M-plane is a plane that contains an M-line and the straight line perpendicular

to both the M-line and linkl3axis M-plane is thus undetermined only if theM-line is collinear withl3axis This special case will present no difficulty:Here motion planning is trivial and amounts to changing only values l3;hence we will disregard this case

• V-plane contains the M-line and is parallel to link l3axis

For our Cartesian arm, the M-line, M-plane, and V-plane map inC-space into a

straight line and two planes, respectively

Figure 6.5 C-space and C-obstacles that correspond to W -space in Figures 6.2 and

6.4 Thicker dotted and solid lines show intersections between obstacles Shown also are projections of the three obstacles on the planel1, l2

Local Directions. Similar to other algorithms in previous chapters, a common

operation in the algorithm here will be the choice of a local direction for the

next turn (say, left or right) This will be needed when, while moving along acurve, the C-point encounters a sort of T-intersection with another curve (which

is here the horizontal part of “T”) Let us define the vector of current motion p

and consider all possible cases

1 TheC-point moves along the M-line or along an intersection curve between

the M-plane and an obstacle and is about to leave M-plane at the

cross-point Define the normal vector m of the M-plane [97] Then the local

direction b is upward if b · m > 0 and downward if b · m ≤ 0.

2 TheC-point moves along the M-line or along an intersection curve between

the V-plane and an obstacle, and it is about to leave V-plane at the point Let l3 be the vector of l3 axis Then, local direction b is left if

cross-b· (p × l3) > 0 and right if b · (p × l3)≤ 0

3 In a special case of motion along the M-line, the directions are ST =

forward and  T S = backward.

Consider the motion of aC-point in the M-plane When, while moving along

the M-line, theC-point encounters an obstacle, it may define on it a hit point , H

Here it has two choices for following the intersection curve between the M-planeand the obstacle surface: Looking from S toward T , the direction of turn at H

is either left or right We will see that sometimes the algorithm may replace the

current local direction by its opposite When, while moving along the intersectioncurve in the M-plane, theC-point encounters the M-line again at a certain point,

it defines here the leave point , L Similarly, when the C-point moves along a

V-plane, the local directions are defined as “upward” and “downward,” where

“upward” is associated with the positive and “downward”—with the negativedirection ofl3axis

6.2.2 The Approach

Similar to other cases of sensor-based motion planning considered so far, ceptually we will treat the problem at hand as one of moving a point automaton

con-in the correspondcon-ingspace (This does not mean at all, as we will see, that

C-space needs to be computed explicitly.) Essential in this process will be sensinginformation about interaction between the arm and obstacles, if any This infor-mation—namely, what link and what part (front or rear) of the link is currently

in contact with an obstacle—is obviously available only in the workspace.Our motion planning algorithm exploits some special topological character-istics of obstacles in C-space that are a function of the arm kinematics Note

that because linksl1,l2, and l3are connected sequentially, the actual number ofdegrees of freedom available to them vary from link to link For example, linkl1

has only one degree of freedom: If it encounters an obstacle at some valuel
1, itsimply cannot proceed any further This means that the correspondingC-obstacle

occupies all the volume ofC-space that lies between the value l1
and one of thejoint limits of joint J1 This C-obstacle thus has a simple structure: It allows

the algorithm to make motion planning decisions based on the simple fact of alocal contact and without resorting to any global information about the obstacle

in question

A similar analysis will show thatC-obstacles formed by interaction between

linkl2and obstacles always extend in C-space in the direction of one semi-axis

of linkl2and both semi-axes of linkl3; it will also show thatC-obstacles formed

by interaction between linkl3and obstacles present generalized cylindrical holes

inC-space whose axes are parallel to the axis l3 No such holes can appear, forexample, along the axesl1orl2 In other words, C-space exhibits an anisotropy

property; some of its characteristics vary from one direction to the other more, C-space possesses a certain property of monotonicity (see below), whose

Further-effect is that, no matter what the geometry of physical obstacles inW -space, no

holes or cavities can appear in aC-obstacle.

From the standpoint of motion planning, the importance of these facts is inthat the local information from the arm’s contacts with obstacles allow one toinfer some global characteristics of the correspondingC-obstacle that help avoid

directions of motion leading to dead ends and thus avoid an exhaustive search.Whereas the resulting path planning algorithm is used in the workspace, with-out computations ofC-space, it can be conveniently sketched in terms of C-space,

as follows If theC-point meets no obstacles on its way, it will move along the

M-line, and with no complications the robot will happily arrive at the targetposition T If the C-point does encounter an obstacle, it will start moving along

the intersection curve between the obstacle and one of the planes, M-plane orV-plane The on-line computation of points along the intersection curve is easy:

It uses the plane’s equation and local information from the arm sensors

If during this motion theC-point meets the M-line again at a point that satisfies

some additional condition, it will resume its motion along the M-line Otherwise,theC-point may arrive at an intersection between two obstacles, a position that

corresponds to two links or both front and rear parts of the same link contactingobstacles Here the C-point can choose either to move along the intersection

curve between the plane and one of the obstacles, or move along the intersectioncurve between the two obstacles The latter intersection curve may lead the C-

point to a wall, a position that corresponds to one or more joint limits In thiscase, depending on the information accumulated so far, theC-point will conclude

(correctly) either that the target is not reachable or that the direction it had chosen

to follow the intersection curve would lead to a dead end, in which case it willtake a corrective action

At any moment of the arm motion, the path of theC-point will be constrained

to one of three types of curves, thus reducing the problem of three-dimensionalmotion planning to the much simpler linear planning:

• The M-line

• An intersection curve between a specially chosen plane and the surface of

aC-obstacle

• An intersection curve between the surfaces of twoC-obstacles

To ensure convergence, we will have to show that a finite combination of suchpath segments is sufficient for reaching the target position or concluding that thetarget cannot be reached The resulting path presents a three-dimensional curve

inC-space No attempt will be made to reconstruct the whole or part of the space

before or during the motion

Since the path planning procedure is claimed to converge in finite time, thismeans that never, not even in the worst case, will the generated path amount to

an exhaustive search

An integral part of the algorithm is the basic procedure from the Bug familythat we considered in Section 3.3 for two-dimensional motion planning for apoint automaton We will use, in particular, the Bug2 procedure, but any otherconvergent procedure can be used as well

6.2.3 Topology of W-Obstacles and C-Obstacles

Monotonicity Property. Obstacles that intersect the W -space volume may

interact with the arm during its motion As mentioned above, one result of suchinteraction is the formation of obstacle shadows Consider the spherical obsta-cle O1 in Figure 6.2 Clearly, no points directly above O1 can be reached byany point of the arm body Similarly, no point of W -space below the obstacle

O2 or to the left of the cubical obstacle O3 can be reached Subsequently, thecorresponding W -obstacles become as shown in Figure 6.4, and their C-space

representation becomes as in Figure 6.5 This effect, studied in detail below, iscaused by the constraints imposed by the arm kinematics on its interaction withobstacles Anisotropic characteristics ofW -space and C-space present themselves

in a special topology ofW - and C-obstacles best described by the notion of the

(W - and C-) obstacle monotonicity:

Obstacle Monotonicity In all cases of the arm interference with an obstacle, there is at least one direction corresponding to one of the axes l i , i = 1, 2, 3,

such that if a value l i
of link l i cannot be reached due to the interference with

an obstacle, then no value l i

> l
i in case of contact with the link front part, or, inversely, l i

< l
i in case of contact with the link rear part, can be reached either.

In what follows, most of the analysis of obstacle characteristics is done interms ofC-space, although it applies to W -space as well Comparing Figures 6.2

and 6.5, note that although physical obstacles occupy a relatively little part ofthe arm’s workspace, their interference with the arm motion can reduce, oftendramatically, the volume of points reachable by the arm end effector The kine-matic constraints are due to the arm joints, acting differently for different jointtypes, and to the fact that arm links are connected in series As a result, thearm effectively has only one degree of freedom for control of motion of linkl1,two degrees of freedom for control of linkl2, and three degrees of freedom forcontrol of linkl3 A simple example was mentioned above on how this can affectpath planning: If during the arm motion along M-line the linkl1hits an obstacle,then, clearly, the task cannot be accomplished

The monotonicity property implies that C-obstacles, though not necessarily

convex, have a very simple structure This special topology of W - and

C-obstacles will be factored into the algorithm; it allows us, based on a givenlocal information about the arm interaction with the obstacle, to predict impor-tant properties of the (otherwise unknown) obstacle beyond the contact point.The monotonicity property can be expressed in terms more amenable to the pathplanning problem, as follows:

Corollary 6.2.1 No holes or cavities are possible in a C-obstacle.

W -obstacle monotonicity affects differently different links and even

differ-ent parts—front or rear—of the same link This brings about more specializednotions ofl i -front and l i -rear monotonicity for every link, i = 1, 2, 3 (see more

below) By treating links’ interaction with obstacles individually and by makinguse of the information on what specific part—front or rear—of a given link iscurrently in contact with obstacles, the path planning algorithm takes advantage

of the obstacle monotonicity property Because this information is not available

inC-space, the following holds:

Information Loss due to Space Transition Information is lost in the space sition W → C Since some of this information—namely, the location of contact points between the robot arm and obstacles—is essential for the sensor-based planning algorithm, from time to time the algorithm may need to utilize some information specific to W -space only.

tran-We will now consider some elemental planar interactions of arm links withobstacles, and we will show that if a path from start to target does exist, then acombination of elemental motions can produce such a path Define the following:

• Type I obstacle corresponds to a W - or C-obstacle that results from the

interaction of linkl1with a physical obstacle

• Type II obstacle corresponds to a W - or C-obstacle that results from the

interaction of linkl2with a physical obstacle

• Type III obstacle corresponds to a W - or C-obstacle that results from the

interaction of linkl3with a physical obstacle

We will use subscripts “+” and “−” to further distinguish between obstacles thatinteract with the front and rear part of a link, respectively For example, a Type

III+ obstacle refers to aC-obstacle produced by interaction of the front part of

linkl3with some physical obstacle

In the next section we will analyze separately the interaction of each linkwith obstacles Each time, three cases are considered: when an obstacle interactswith the front part, the rear part, or simultaneously with both parts of the link

in question We will also consider the interaction of a combination of links withobstacles, setting the foundation for the algorithm design

Interaction of Link l1with Obstacles— Type I Obstacles. Since, according

to our model, sliding along an obstacle does not constitute an interference withthe link l1 motion, we need to consider only those cases where the link meets

an obstacle head-on When only the front end of link l1 is in contact with anobstacle—say, at the joint value l1
—a Type I+ obstacle is produced, which

extends from C-space floor to ceiling and side to side (see Figure 6.6) which

effectively reduces theC-space cubicle by the volume (l1 max− l

Formally the property of Type I obstacle monotonicity is expressed as follows:

Figure 6.6 C-space with a Type I obstacle.

Type I Monotonicity For any obstacle interacting with link l1, there are three directions corresponding to the joint axes l i , i = 1, 2, 3, respectively, along which

the C-obstacle behaves monotonically, as follows: If a position (l1
, l
2, l3
) cannot

be reached by the arm due to an obstacle interference, then no position (l1

, l

2, l3

), such that l1

> l1
in case of the (obstacle’s) contact with the link’s front part,

or l1

< l1
in case of the contact with the link’s rear part, and l

2∈ [0, l2 max],

l3

∈ [0, l3 max], can be reached either.

Interaction of Link l2with Obstacles— Type II Obstacles

Front Part of Link l2—Type II+ Obstacles. Consider the case when only thefront part of link l2 interferes with an obstacle (Figure 6.2) Because link l2

effectively has two degrees of freedom, the corresponding Type II+obstacle willlook inC-space as shown in Figure 6.7 The monotonicity property in this case

is as follows:

Type II+ Monotonicity For any obstacle interacting with the front part of link

l2, there are two axes (directions), namely l2and l3, along which the C-obstacle behaves monotonically, as follows: If a position (l
1, l2
, l3
) cannot be reached by the arm due to an obstacle interference, then no position (l1
, l2

, l3

), such that

l2

> l2
and l3

∈ [0, l3 max], can be reached either.

As a result, a Type II+collision, as at pointH in Figure 6.7, indicates that any

motion directly upward or downward fromH along the obstacle will necessarily

bring theC-point to one of the side walls of the C-space cubicle This suggests

l2

l1

H O

H ′

a ′

S ′ S

M-planeab

L T

Figure 6.7 (a)W -space and (b) C-space with a Type II obstacle (S, T ) is the M-line; HabL is a part of the intersection curve between the obstacle O and M-plane.

that a plane can be chosen such that the exploration of the intersection curve

between this plane and the Type II+ obstacle will produce a more promisingoutcome that will result either in a success or in the correct conclusion that thetarget cannot be reached In the algorithm, the M-plane will be used, which offerssome technical advantages In general, all three arm joints will participate in thecorresponding motion

For this case (front part of link l2 interacting with an obstacle), the decision

on which local direction, right or left, is to be taken at a hit point H in order

to follow the intersection curve between an M-plane and a Type II+ obstacle ismade in the algorithm based on the following rule:

Rule 1:

If l1H > l1T, the current direction is “left.”

If l1H < l1T, the current direction is “right.”

If l1H = l1T, the target cannot be reached

Rear Part of Link l2—Type II−Obstacles. Now consider the case when only therear part of linkl2—that is, the link’s part to the left of joint J2—can interferewith obstacles (see obstacle O3, Figure 6.2) This situation produces a C-space

very similar to that in Figure 6.7 The direction of obstacle monotonicity alongthe axisl2 will now reverse:

Type II− Monotonicity For any obstacle interacting with the rear part of link

l2, there are two axes (directions), namely l2and l3, along which the C-obstacle behaves monotonically, as follows: If a position (l
, l
, l
) cannot be reached by

the arm due to an obstacle interference, then no position (l1
, l2

, l3

), such that

l2

< l2
and l3

∈ [0, l3max ], can be reached either.

In terms of decision-making, this case is similar to the one above, except thatthe direction of obstacle monotonicity along l2 axis reverses, and the choice ofthe current local direction at a hit pointH obeys a slightly different rule:

Rule 2:

Ifl1H > l1T, the current direction is “right.”

Ifl1H < l1T, the current direction is “left.”

Ifl1H = l1T, the target cannot be reached

Interaction of Both Parts of Link l2with Obstacles. Clearly, when both the frontand the rear parts of linkl2 interact simultaneously with obstacles, the resulting

Type II+ and Type II− obstacles fuse into a single obstacle that divides

C-space into two separate volumes unreachable one from another (see Figure 6.8)

If going fromS to T requires the arm to cross that obstacle, the algorithm will

conclude that the target position cannot be reached

Figure 6.8 C-space in the case when both front and rear parts of link l2 interact with

obstacles, producing a single obstacle that is a combination of a Type II+and Type II−

obstacles.

Stalactites and Stalagmites: Type III Obstacles

Front Part of Link l3—Type III+ Obstacles. Assume for a moment that onlythe front part of link l3 can interfere with an obstacle (see, e.g., obstacle O1,Figures 6.2 and 6.4) Consider the cross sections of the obstacle with two hori-zontal planes: one corresponding to the value l3
and the other corresponding tothe value l3

, with l3
< l3

Denote these cross sections a
and a

, respectively.Each cross section is a closed set limited by a simple closed curve; it may ormay not include points on theC-space boundary Because link l3is a generalizedcylinder, the vertical projection of one cross section onto the other satisfies therelationshipa
⊆ a

This is a direct result of the Type III

+obstacle monotonicityproperty, which is formulated as follows:

Type III+Monotonicity For any obstacle interacting with the front part of link l3, there is one axis (direction), namely l3, along which the corresponding C-obstacle behaves monotonically, as follows: if a position (l
1, l2
, l3
) cannot be reached by the arm due to an obstacle interference, then no position (l1
, l2
, l3

) such that

l

3> l3
can be reached either.

This property results in a special “stalactite” shape of Type III+obstacles Atypical property of icicles and of beautiful natural stalactites that hang down fromthe ceilings of many caves is that their horizontal cross section is continuously

reduced (in theory at least) from its top to its bottom Each Type III+ obstaclebehaves in a similar fashion It forms a “stalactite” that hangs down from theceiling of theC-space cubicle, and its horizontal cross section can only decrease,

with its maximum horizontal cross section being at the ceiling level, l3= l3 max

(see cubicle Oabcdefg and obstacle O1, Figure 6.4) For any two horizontal cross

sections of a Type III+ obstacle, taken at levelsl
3and l3

such that l3

> l3
, theprojection of the first cross section (l3
level) onto a horizontal plane contains nopoints that do not belong to the similar projection of the second cross section (l3

level) This behavior is the reflection of the monotonicity property

Because of this topology of Type III+ obstacles, the sufficient motion formaneuvering around any such obstacle—that is, motion sufficient to guaranteeconvergence—turns out to be motion along the intersection curves between thecorresponding C-obstacle and either the M-plane or the V-plane (specifically,

its part below M-plane), plus possibly some motion in the floor of the C-space

cubicle (Figure 6.9)

Rear Part of Link l3—Type III− Obstacles. A similar argument can be madefor the case when only the rear end of link l3 interacts with an obstacle (see,e.g., obstacle O2, Figures 6.2, 6.4, and 6.5) In C-space the corresponding Type III− obstacle becomes a “stalagmite” growing upward from the C-space floor.

This shape is a direct result of the Type III− obstacle monotonicity property,which is reversed compared to the above situation with the front part of link l3,

as follows: