Sensing Intelligence Motion - How Robots & Humans Move - Vladimir J. Lumelsky Part 9 docx

Assume that the arm’s sensing allows it to sense objects inW -space within some sensing range r Figure 5.16.. The sensing range image will look in C-space more or less like the one in Fi

the sensing range will be a complex figure (Figure 5.16) that surrounds the armand changes its shape as the arm links move relative to each other.

InC-space the situation is somewhat closer to the mobile robot sensing range

simply because in C-space the arm becomes a point But that’s where the

simi-larity stops Assume that the arm’s sensing allows it to sense objects inW -space

within some sensing range r (Figure 5.16) An obstacle will be detected when it

is at a distance equal to or smaller thanr from the point on the arm body closest

to it

With this kind of proximity sensing, the robot’s sensing range in C-space

plane (θ1, θ2) is not a circle anymore In fact, it is not even an entity with fixedparameters The sensing range image will look in C-space more or less like the

one in Figure 5.17 (point C is the arm’s current position) As the arm links moverelative to the arm base and relative to each other, the joint angles (θ1, θ2) changeaccordingly The sensing range C-space image then moves in the plane (θ1, θ2),

Figure 5.17 In C-space the sensing range of the revolute–revolute arm has a shape

similar to the one shown here The point in the center of the figure corresponds to the values (θ1, θ2 ) of the arm’s current position As the arm moves in its workspace, this figure moves inC-space, with its shape and dimensions changing continuously.

PLANAR REVOLUTE–REVOLUTE (RR) ARM 217

with its shape and dimensions shrinking and expanding from rectangle-like toellipse-like, and with shapes in between like in Figure 5.17 Animation of thisprocess makes for a wonderful movie: One sees a strange creature that is movingwhile constantly changing its shape according to some mystifying law The extent

of variability in the sensing rangeC-space image depends on the sensing range

r and the arm’s kinematics.

Calculation of the sensing rangeC-space image is an interesting though rather

involved task; there are many details and many special cases to attend to Withgood equations for the sensing range, one could improve motion planning algo-rithms by providing a look-ahead optimization of the arm’s next few steps, orattempt algorithms that take into account the arm dynamics, similar to the work

we did in Chapter 4 To my knowledge, today there are no published analyses

on this topic As a first approximation, one can start with a simplified model ofthe sensing range, presenting it as a circle whose radius changes as a function

of the arm position (θ1, θ2) A conservative approximation would be to modelthe arm sensing by the maximum circle inscribed in the real sensing range Withthis model the robot would be safe, but much sensing would be wasted: In somedirections in the (θ1, θ2) plane the actual sensing will go much farther than thecircular model will indicate

As the arm moves, its sensing range image inC-space “breathes,” shrinking

and expanding as it moves in the plane (θ1, θ2) The extent of such changesdepends on the motion It is easy to see, for example, that if we fix angleθ2andlet angleθ1change, inC-space of Figure 5.17 the sensing range figure will move

horizontally, and its shape will remain the same This is because the motion doesnot involve any changes in the relative position of linksl1 and l2 Any motioninvolving a change in angle θ2 will cause changes in the shape of the sensingrange figure

Except for the added calculation due to the variable sensing range inC-space,

incorporating proximity sensing in the arm motion planning algorithm is similar

to the analogous process for mobile robots (Section 3.6) One can combine, forexample, one of the VisBug algorithms for a mobile robot (Section 3.6) with theRR-Arm Algorithm developed in this chapter The fact that the latter is noticeablymore complex than Bug algorithms calls for a careful analysis To date, thereare no published results in this area, in spite of its significant theoretical andpractical potential

How proximity sensing can affect the RR-Arm Algorithm performance can

be seen in Figure 5.18 Here linkl2 happens to be attached to linkl1 not by itsendpoint, as in some of our prior figures, but by some other point on the link.(This is a more realistic design; it often occurs in industrial arm manipulators.)Note how elegant and economical the arm’s path becomes when the arm is pro-vided with proximity sensing (Figure 5.18b), compared to its performance withtactile sensing (Figure 5.18a) In fact, the robot path in Figure 5.18b is almostthe optimal path between the S and T locations; it could hardly be improved

even by a procedure operating with complete information This of course will

(b)

Figure 5.18 Performance of RR-Arm Algorithm in a scene with four obstacles (black

objects): (a) with tactile sensing and (b) with proximity sensing.

not be always so; as we know, obtaining information about the scene “on the fly”rather than beforehand precludes one from guaranteeing the optimal solution.4

kine-they also allowed us to reduce the problem from searching the whole space

to searching only a tiny one-dimensional subset of space Analysis carriedout in this section sheds much light on the motion planning issues involved,and it should serve us well in studying other arm configurations in this andnext chapters

• The fact that the C-space of the RR arm is a two-dimensional manifold,

namely a common torus, turned out to play an important role in the RR-Arm

4 As we will see in Chapter 7, even with the benefit of seeing the whole scene and of prior training, humans are not able to compete even with the performance shown in Figure 5.18a, let alone with that in Figure 5.18b This fact is at the heart of the argument for synergistic human–robot systems, where responsibilities between the partners are divided in accordance with their abilities.

PLANAR REVOLUTE–REVOLUTE (RR) ARM 219

Algorithm, not lastly because thisC-space structure allows for more than

one “short” route between the start and target positions, which have beenused with profit by the algorithm The analysis demonstrates that the armkinematics can greatly influence the algorithm structure In the followingsections we will study in a similar vein the remaining four of the five con-figurations of planar two-link arms shown in Figure 5.1 We will concludethese studies with an attempt, in Section 5.8.4, to develop a unifying theorythat will allow one to consider each of the five kinematics of Figure 5.1 as

a special instant of one general case

• Planning of arm motion with the described RR-Arm Algorithm is donecompletely in the workspace (W -space), based on the sensing data from

the arm sensors The above analysis and examples referring to the armconfiguration space have been used solely to establish the theory and developthe algorithmic machinery

• A similar consideration applies to the sensing used Whereas most of ouralgorithm design process relied on tactile sensing, this was done only forthe sake of simpler explanation As discussed in Section 5.2.5 (and more

in Chapter 8), proximity sensing, and not tactile sensing, should be used inpractical arm manipulator motion planning systems

• No preliminary exploration of obstacles and no beforehand partial or plete computation of the scene in W -space or C-space takes place or is

com-expected by the algorithm By the time the arm arrives at the target location,

it may know very little about the space that it just traversed

• If the desired target position is not feasible because of interfering cles, the reachability test built into the algorithm will make this conclusion,usually quickly enough and without exploring the whole space

obsta-• The algorithm plans the robot arm motion better than humans do We willdiscuss this interesting observation in great detail in Chapter 7 In brief,when watching the RR-Arm Algorithm in action, humans have difficultygrasping its mechanism or the rationale behind the paths it generates Aquick glance at the paths in Figures 5.14, 5.15, and 5.18 should help con-vince one that this is so This is so even for simple scenes, and it is so fortactile as well as for more complex sensing The difficulty for humans is not

in that the algorithm is overly complex With quick training, one will be able

to understand and use the RR-Arm Algorithm in Cspace—but not in W

-space Unfortunately, this would be a useless demonstration becauseC-space

is not available for motion planning; remember, our primary assumption isthat no information about the scene is available beforehand On the otherhand, asking human operators to use the algorithm in the workspace, theway a robot arm manipulator does it, turns out to be hopeless And humansown strategies, whatever they are, consistently show an inferior performancecompared to that of RR-Arm Algorithm (see Chapter 7)

Recall how very different our current situation is from the one we faced withmobile robot motion planning algorithms (Chapter 3) We observed there that,

first, humans’ own motion planning strategies work pretty well in the relatedtasks and, second, humans have no problem interpreting, learning, and usingrelevant motion planning algorithms In comparison, motion planning for even

a simple arm manipulator is a task that poses serious mental challenges to ahuman This fact goes a long way in explaining difficulties that human operatorsexhibit when controlling real-world teleoperated robotics systems The price forthose difficulties is operators’ mistakes and an overly slow operation that rulesout many real-life tasks

This suggests the need for changes in today’s approaches to the design ofteleoperated systems In particular, it is highly desirable to shift the responsibil-ity for obstacle collision avoidance from the operator’s shoulders to the robotintelligence We will return to this topic in Chapters 7 and 8

5.3 DISTINCT KINEMATIC CONFIGURATIONS OF RR ARM

Even for the most popular revolute and sliding (prismatic) joint types, eachcombination of joint types of an arm manipulator can be realized in more thanone kinematic configuration The RR arm that we analyzed above (Section 5.2)

is especially prolific in terms of variability of kinematic configurations We willreview here those configurations, as an example of how such variability occurs

as well as to see how the theory developed above applies to them This exercisealso helps train one’s spatial intuition, a useful quality in the work we do here.Four different configurations of RR arms are shown in Figure 5.19 As wewill see, different kinematic configurations result in different, sometimes quiteunusual, configurations of their workspace This fact does not change the basics

of sensor-based motion planning considered above No matter how an RR arm

is configured, the following statements are true:

(a) The arm’s two degrees of freedom guarantee that its endpoint moves in atwo-dimensional manifold, be it a plane or some other surface

(b) The arm’s configuration space is still a common torus, and so the theoryand the motion planning algorithm developed in Section 5.2 applies

RR Arm of Figure 5.19a. This arm is recognized easily—it is the same planartwo-link RR arm manipulator that has been studied in much detail in Section 5.2(see Figures 5.1a and 5.2) The arm lies and moves in the plane Its two jointaxes are parallel to each other and perpendicular to the arm’s plane The maindifference between this arm and the remaining three arms in Figure 5.19 is thatthe two joint axes of those three arms are mutually perpendicular rather thanparallel.5 This changes the arm workspace rather dramatically

5 There exist arm designs where joint axes intersect at angles different from parallel or perpendicular Some such designs have even been patented, because they provide interesting kinematic properties.

No such kinematics is considered in this text.

DISTINCT KINEMATIC CONFIGURATIONS OF RR ARM 221

Figure 5.19 Four different kinematic configurations of the RR (revolute– revolute) arm:

(a) RR arm studied in detail in Section 5.2; the arm’s endpoint moves in the plane (b) This arm’s endpoint moves on the surface of a sphere (c) This arm’s endpoint moves

on the surface of a torus (d) This arm’s endpoint moves on the surface of a truncated

sphere.

RR Arm of Figure 5.19b. See Figure 5.20a, which shows both revolute joints

of this arm,J1 andJ2, in the same spot, with the joint axes intersecting at 90◦.Because link l1 produces no physical displacement, we can take it as being ofzero length, l1= 0 Then, only one link, l2, is physically present in this arm;

hence the arm looks like an outstretched human arm Sometimes this mechanism

is interpreted as a single ball joint Since from the control standpoint this device

still has two independent control variables, seeing it as two independent revolutejoints, rather than one (ball) joint, is more in line with our other notation inthis text

The arm’s first joint angle,θ1, is responsible for motion in one plane; for ficity, assume this is a horizontal plane The second joint angle,θ2, is responsiblefor the arm motion in the vertical plane Together they allow the arm endpoint(end effector) to reach any point on the surface of a sphere of radius l2 Theworkspace (W -space) of this arm is hence a sphere Any point P on the sphere

speci-corresponds to two joint solutions, (θ1P , θ2P ) and (π + θ P

Proceeding in this direction, we want to chose an M-line, the line that the armendpoint would go through if no obstacle interfered with the arm motion Sincethe C-space (configuration space) of this arm is a torus, the choice is among

four M-lines (Section 5.2) These are shown in Figure 5.20 Denote the positivedirection of change of angleθ i , i = 1, 2, by “+” and denote the negative one by

“−” Then the four M-lines are four geodesic curves, as shown in Figure 5.20b,with the corresponding joint angles changing as follows:

accord-RR Arm of Figure 5.19c. A detailed picture of this arm configuration is shown

in Figure 5.21a The only difference between this configuration and the one inFigure 5.20a is that here the arm’s two joints are at a distance from each other,equal to the length of the first link,l1 Linksl1 andl2lie in the same plane—ingeneral, depending on linkl1shape, in parallel planes The arm’s endpoint movesalong the surface of a torus, and so itsW -space is a torus.∗This torus may or maynot have a hole depending on the relation between the lengths of linksl1andl2:

l2> l1produces aW -space torus with no hole;

l2< l1produces aW -space torus with a hole.

Projections ofW -space onto the horizontal (xy) and vertical (xz or yz) planes for

both cases, l2> l1 and l2< l1, are shown in Figures 5.21b and 5.21c, tively

respec-∗To emphasize, it is not the configuration space that forms a torus here, as we had it before, but the

workspace.

DISTINCT KINEMATIC CONFIGURATIONS OF RR ARM 223

z

y T

x

S b

Figure 5.20 The RR arm of Figure 5.19b (a) The arm design (b) The arm’s workspace,

with four complementary M-lines shown (The sketches are not to the same scale.)

Figure 5.21 The RR arm of Figure 5.19c (a) The arm design (b) Two projections of

the armW -space in case l2> l1 (c) Two projections of the armW -space in case l2< l1

in Figure 5.22a The difference between this configuration and the one inFigure 5.21c is that here linkl2rotates in the plane perpendicular, rather than par-allel, to linkl1 In other words, the arm looks like a fan, with linkl2being the fanpropeller’s only blade This design makes for a somewhat strange workspace: Thearm’s endpoint moves along the surface of a truncated sphere of radius



l2

1+ l2 2

DISTINCT KINEMATIC CONFIGURATIONS OF RR ARM 225

Figure 5.22 The RR arm of Figure 5.19d (a) Arm design (b) ArmW -space.

(Figure 5.22b) Linkl1moves in the horizontal planexy Link l2is always insidethe volumeD limited by the sphere and a cylinder whose radius is l1and whoseheight is 2l2(see Figure 5.22b; the cross section of the volumeD is shaded) The

body of linkl2 may therefore interact with 3D obstacles that happen to appear

in volumeD.

5.4 PRISMATIC – PRISMATIC (PP, OR CARTESIAN) ARM

This arm is the second one, arm (b), among the five arms shown in Figure 5.1.The reason it is called Cartesian is that the displacement in each of its jointstranslates directly in exactly the same motion of the arm endpoint in Cartesianplane (see Figure 5.23) The arm has two prismatic (sliding) joints, with jointvalues l1andl2—hence its other name, a PP Arm The boundaries of the arm’s

W -space are limited by the rectangle whose sides are equal to the maximum

lengths of links l1 and l2 We assume that no obstacles outside W -space can

interfere with the arm motion; hence the path planning problem is limited tothe arm’s W -space rectangle The M-line is defined as a straight-line segment

connecting the arm’s starting and target points, S and T

From Section 5.2.1, a shadow of an obstacle X is the area of workspace no

point of which can be reached by the arm endpoint due to interference of Xwith the arm motion Functionally, then, for the arm an obstacle shadow is thesame as the physical obstacle that causes it Observe in Figure 5.23 that for anyobstacle in W -space of the PP arm, no points to the right of an obstacle can

ever be reached by the arm endpoint We will see that this property of this armkinematics makes motion planning a rather simple task

The boundary of an obstacle plus the boundary of the shadow that it forms

produces the virtual line of the obstacle The said property of the PP arm—that

no points to the right of an obstacle can be reached by the arm endpoint—means

l 2 max

l 1 max a

b

0

3 4

T S

A

Figure 5.23 Cartesian (PP) arm The virtual obstacle includes the actual obstacle and its shadow (shaded).

PRISMATIC–PRISMATIC (PP, OR CARTESIAN) ARM 227

that any obstacle’s virtual line is a simple curve In Figure 5.23, the virtual line

of the circular obstacle A is the line that passes through points 1, 2, 3, 4, 5, 6, 7.Recall that in the arm configuration space (C-space), the virtual obstacle is the C-space image of the corresponding physical obstacle The virtual boundary of

the obstacle is theC-space image of the corresponding virtual line In Figure 5.23,

the virtual boundary of obstacle A is the same line (1, 2, 3, 4, 5, 6, 7) Clearly,this will be always so Hence, any virtual boundary is a simple curve This simplecorrespondence makes the PP arm’sC-space practically identical to its W -space,

and hence unnecessary for further analysis

As we will see later, the above simple structure of PP arm virtual obstaclesbrings about one specific condition that allows a direct and simplified use forthis arm of the basic planning procedure developed in Section 3.3, and ensuresits convergence

Another condition that the basic planning procedure requires—that the virtualboundary be a closed curve—does not hold for the PP arm For example, inFigure 5.23 no point of the line segment between points 7 and 1 of the obstacleshadow can be reached by the arm endpoint, and so it is not a part of the virtualboundary This will present no difficulty for the planning procedure, though Ingeneral, an obstacle’s virtual boundary in C-space of this arm consists of four

distinct segments:

• The “left” curve corresponding to the arm endpoint’s following the actualobstacle boundary; in Figure 5.23 this segment comprises points 3-4-5

• Two mutually parallel straight line segments corresponding to those points

of the arm body (other than the arm endpoint) that touch the actual obstacle;

in Figure 5.23 these are lines 3-2-1 and 5-6-7

• The straight-line segment that is a part of theW -space boundary; in

Fig-ure 5.23, this is line 7-1

Of these, the first three segments form a simple open curve, each point of whichcan be reached by the arm endpoint The fact that the fourth segment cannot bereached by the arm endpoint poses no algorithmic problems since the endpoints

of the said simple open curve—in Figure 5.23, points 1 and 7—can easily berecognized from the fact that they always lie on theW -space boundary and hence

correspond to the maximum value of the joint value l2 This fact will help inshowing the algorithm convergence An important statement that helps simplifythe path planning procedure of the PP arm follows directly from Figure 5.23:

Lemma 5.4.1 For the Cartesian two-link arm, if the target point T is able from the starting point S, then there exists a path from S to T such that it corresponds to a monotonic change of the joint value l1.

reach-This can be shown as follows Depending on whether the difference (l S

1 − l T

1)

is positive, zero, or negative, establish the direction of change of link l1 motionalong the M-line from S to T —positive, zero, or negative, respectively Note

that this information is known before the motion takes place, from coordinates

of points S and T

If during its motion along the M-line from S to T the arm encounters an

obstacle, such a local direction is chosen for passing around the obstacle forwhich the corresponding change in joint valuel1coincides with the establishedM-line direction of change of l1 In the special case ofl1 being constant in thevicinity of the hit point, the local direction should be chosen as corresponding

to decreasing values ofl2; otherwise the arm will not be able to pass around theobstacle

If, while passing around an obstacle, the current value l1 moves outside ofthe interval (l1S , l T

1), then, clearly, pointT lies in the shadow of the obstacle and

cannot be reached If the M-line direction of change ofl1is “0”—which will be

so if the M-line happen to be parallel to link l2—and an obstacle is met alongthe way, then, again, point T cannot be reached because it is in the shadow

of the obstacle In other words, Lemma 5.4.1 holds, and this helps simplify theplanning procedure

While the PP arm planning procedure will work for the arm links of anyshapes, it can be further simplified and made more efficient if linkl2is assumed

to present an elongated rectangle whose sides are parallel to the joint axesl1and

l2, respectively If linkl2happens to be of a more complex shape, the algorithmcan replace it with a minimum rectangle that contains the link Hence link l2 inthe example of Figure 5.23 would be treated as a rectangle of width zero

Now the whole path planning procedure for the PP-Arm Algorithm can be

(a) Target T is reached The procedure stops.

(b) An obstacle is encountered and a hit point,H j, is defined Choose thelocal direction using the M-line direction of change ofl1 Go to Step 3.

3 The arm follows the obstacle virtual boundary until one of the followingoccurs:

(a) The target T is reached The procedure stops.

(b) The M-line is met at a distanced from T such that d < d(H j , T ); point

L j is defined Incrementj Go to Step 2.

(c) The current valuel1is outside of the interval (l1S , l T

1) Target T cannot

be reached The procedure stops

It can be shown that the algorithm will work correctly if the arm links andobstacles in the scene are of arbitrary chapes

REVOLUTE–PRISMATIC (RP) ARM WITH PARALLEL LINKS 229

5.5 REVOLUTE – PRISMATIC (RP) ARM WITH PARALLEL LINKS

The revolute–prismatic parallel kinematic configuration (Figure 5.1c) is a mon major linkage in commercial robot arm manipulators The so-called Stanfordmanipulator robot (see, e.g., Ref 7) is a typical example of this type Joint values

com-of this arm are the angle θ1 of the first (revolute) joint and the variable length

l2 of the second (prismatic) joint; 0≤ l2 ≤ l2 max (see Figure 5.24a) The outerboundary of the arm’s workspace (W -space) is a circle of radius (l1+ l2 max) Its

inner boundary is a circle of radiusl1, which defines the dead zone inaccessible

to the arm endpoint Unlike in Figure 5.1c, in order to not overcrowd the picture,

no dead zone appears in the arm in Figure 5.24a; that is, here l1= 0 With thearm endpointb in some position of W -space—say, S —the position of the link’s

rear end, a S, can be found by passing a line segment of length l2 max from b S

through the originO.

For specificity, let us define the M-line as a straight-line segment connectingpoints S and T ; denote it M1-line In the example in Figure 5.24a, one pathfromS to T would be as shown, the curve (S, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, T ).

Observe that if due to obstacles the arm cannot reach pointT by following the

M1-line, it might be able to reach T “from the other side,” by changing angle

θ1 in the direction opposite to that of the M1-line Hence, similar to how we

did it with the RR arm (Section 5.2), a complementary M-line, the M2-line, isintroduced, defined as consisting of three parts: two straight-line segments, (S, S
)and (T , T
), continuing theM1-line segment outwards until its intersection withtheW -space outer boundary, and a segment of the W -space outer boundary that

corresponds to the interval ofθ1complementing that of theM1-line to 2π This

choice for the complementary M-line is largely arbitrary; anyM2-line will do, aslong as it is uniquely defined, is computationally simple, and complements the

θ1interval ofM1-line to 2π

As the arm moves along the M-line, obstacles will interfere with its motion

in two different ways In our example the arm endpoint will be forced to leaveM-line two times: The first time is when obstacle A interferes with the rear part

of linkl2, creating the curve (1, 2, 3, 4, 5), (Figure 5.24a), and the second time

is when obstacle B interferes with the front part of link l2, creating the curve(6, 7, 8, 9, 10) (Figure 5.24a) Note two shadows formed during this process(shaded in Figure 5.24a): Under no circumstances the arm endpoint will be able

to reach any point within the figures with boundaries (0, 1, 2, 3, 4, 5, 0) and (6, 7, 8, 9, 10, 11, 12).

Define a front contact of link l2as a situation where a part of the link betweenits front end (which is the part containing the arm endpoint) and the origin is in

contact with the obstacle A rear contact of the link refers to a situation where a

part of the link between its rear end and the origin is in contact with the obstacle

Correspondingly, a front contact forms the front shadow of the obstacle, and a rear contact forms the rear shadow In Figure 5.24a, the rear shadow of obstacle

A is limited by the line(1, 0, 5, 4, 3, 2, 1), and the front shadow of obstacle B is

limited by the line(6, 7, 13, 9, 10, 11, 12, 6).

l2max

q1 = 0

q1 = 90°

S ′ S

1

6

8 9

10 11

4 5 7 6 8

9

10

5 7

Figure 5.24 Revolute– prismatic (RP) parallel two-link arm (a) W -space, with obstacles

A and B Shadows (shaded) result from interaction between the rear end of linkl2 with obstacle A and the front end ofl2with obstacle B (b)C-space images of the corresponding

virtual obstacles Line(S, T ) is the image of the straight line (S, T ) in W -space.