216 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS the sensing range will be a complex figure (Figure 5.16) that surrounds the arm and changes its shape as the arm links move relative to each other. In C-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 in W -space within some sensing range r (Figure 5.16). An obstacle will be detected when it is at a distance equal to or smaller than r 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 (θ 1 ,θ 2 ) is not a circle anymore. In fact, it is not even an entity with fixed parameters. 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 move relative to the arm base and relative to each other, the joint angles (θ 1 ,θ 2 ) change accordingly. The sensing range C-space image then moves in the plane (θ 1 ,θ 2 ), 2p q 2 q 1 q 2 C q 1 C 2p C 0 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 (θ 1 ,θ 2 ) of the arm’s current position. As the arm moves in its workspace, this figure moves in C-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 to ellipse-like, and with shapes in between like in Figure 5.17. Animation of this process makes for a wonderful movie: One sees a strange creature that is moving while constantly changing its shape according to some mystifying law. The extent of variability in the sensing range C-space image depends on the sensing range r and the arm’s kinematics. Calculation of the sensing range C-space image is an interesting though rather involved task; there are many details and many special cases to attend to. With good 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, or attempt 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 of the sensing range, presenting it as a circle whose radius changes as a function of the arm position (θ 1 ,θ 2 ). A conservative approximation would be to model the arm sensing by the maximum circle inscribed in the real sensing range. With this model the robot would be safe, but much sensing would be wasted: In some directions in the (θ 1 ,θ 2 ) plane the actual sensing will go much farther than the circular model will indicate. As the arm moves, its sensing range image in C-space “breathes,” shrinking and expanding as it moves in the plane (θ 1 ,θ 2 ). The extent of such changes depends on the motion. It is easy to see, for example, that if we fix angle θ 2 and let angle θ 1 change, in C-space of Figure 5.17 the sensing range figure will move horizontally, and its shape will remain the same. This is because the motion does not involve any changes in the relative position of links l 1 and l 2 . Any motion involving a change in angle θ 2 will cause changes in the shape of the sensing range figure. Except for the added calculation due to the variable sensing range in C-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, for example, one of the VisBug algorithms for a mobile robot (Section 3.6) with the RR-Arm Algorithm developed in this chapter. The fact that the latter is noticeably more complex than Bug algorithms calls for a careful analysis. To date, there are no published results in this area, in spite of its significant theoretical and practical potential. How proximity sensing can affect the RR-Arm Algorithm performance can be seen in Figure 5.18. Here link l 2 happens to be attached to link l 1 not by its endpoint, 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 with tactile sensing (Figure 5.18a). In fact, the robot path in Figure 5.18b is almost the 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 218 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS ( a ) ST ST ( 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 5.2.6 Concluding Remarks Let us summarize here some properties of the sensor-based motion planning strategy for the revolute–revolute arm manipulator developed in this section. • Of a pivotal importance during the development of RR-Arm Algorithm were the topological properties of the arm configuration space (C-space). These properties not only allowed us to convert the problem from moving a kine- matic jointed arm to moving a point “robot” in the corresponding C-space, they also allowed us to reduce the problem from searching the whole space to searching only a tiny one-dimensional subset of space. Analysis carried out 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 and next 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 this C-space structure allows for more than one “short” route between the start and target positions, which have been used with profit by the algorithm. The analysis demonstrates that the arm kinematics can greatly influence the algorithm structure. In the following sections 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 conclude these studies with an attempt, in Section 5.8.4, to develop a unifying theory that 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 done completely in the workspace (W -space), based on the sensing data from the arm sensors. The above analysis and examples referring to the arm configuration space have been used solely to establish the theory and develop the algorithmic machinery. • A similar consideration applies to the sensing used. Whereas most of our algorithm design process relied on tactile sensing, this was done only for the 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 in practical arm manipulator motion planning systems. • No preliminary exploration of obstacles and no beforehand partial or com- plete computation of the scene in W-space or C-space takes place or is 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 obsta- cles, the reachability test built into the algorithm will make this conclusion, usually quickly enough and without exploring the whole space. • The algorithm plans the robot arm motion better than humans do. We will discuss this interesting observation in great detail in Chapter 7. In brief, when watching the RR-Arm Algorithm in action, humans have difficulty grasping its mechanism or the rationale behind the paths it generates. A quick 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 for tactile 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 C-space—but not in W - space. Unfortunately, this would be a useless demonstration because C-space is not available for motion planning; remember, our primary assumption is that no information about the scene is available beforehand. On the other hand, asking human operators to use the algorithm in the workspace, the way a robot arm manipulator does it, turns out to be hopeless. And humans own strategies, whatever they are, consistently show an inferior performance compared to that of RR-Arm Algorithm (see Chapter 7). Recall how very different our current situation is from the one we faced with mobile robot motion planning algorithms (Chapter 3). We observed there that, 220 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS first, humans’ own motion planning strategies work pretty well in the related tasks and, second, humans have no problem interpreting, learning, and using relevant motion planning algorithms. In comparison, motion planning for even a simple arm manipulator is a task that poses serious mental challenges to a human. This fact goes a long way in explaining difficulties that human operators exhibit when controlling real-world teleoperated robotics systems. The price for those difficulties is operators’ mistakes and an overly slow operation that rules out many real-life tasks. This suggests the need for changes in today’s approaches to the design of teleoperated systems. In particular, it is highly desirable to shift the responsibil- ity for obstacle collision avoidance from the operator’s shoulders to the robot intelligence. 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, each combination of joint types of an arm manipulator can be realized in more than one kinematic configuration. The RR arm that we analyzed above (Section 5.2) is especially prolific in terms of variability of kinematic configurations. We will review 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 exercise also 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 we will see, different kinematic configurations result in different, sometimes quite unusual, 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 a two-dimensional manifold, be it a plane or some other surface. (b) The arm’s configuration space is still a common torus, and so the theory and 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 planar two-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 joint axes are parallel to each other and perpendicular to the arm’s plane. The main difference between this arm and the remaining three arms in Figure 5.19 is that the two joint axes of those three arms are mutually perpendicular rather than parallel. 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 x 2 y o J 1 x o z o (a)(b) (c)(d) J 1 J 2 z o x o x 1 x 2 a 2 J 2 z 2 a 1 x o x 1 z o y o z 1 x 1 q 1 q 2 J 1 x 2 z 1 z 2 q 1 q 2 y o P P a 1 a 2 l 1 l 2 l 1 l 2 y 1 x 1 l 2 l 1 z 1 a 2 J 2 x 2 x 1 y o J 1 x 1 x o x 2 z o P z 1 J 2 l 2 l 1 q 2 q 2 q 1 q 1 P 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, J 1 and J 2 , in the same spot, with the joint axes intersecting at 90 ◦ . Because link l 1 produces no physical displacement, we can take it as being of zero length, l 1 = 0. Then, only one link, l 2 , 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 revolute joints, rather than one (ball) joint, is more in line with our other notation in this text. 222 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS The arm’s first joint angle, θ 1 , is responsible for motion in one plane; for speci- ficity, assume this is a horizontal plane. The second joint angle, θ 2 , is responsible for 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 l 2 .The workspace (W -space) of this arm is hence a sphere. Any point P on the sphere corresponds to two joint solutions, (θ P 1 ,θ P 2 ) and (π +θ P 1 ,π − θ P 2 ). Since the body of link l 2 moves in the three-dimensional (3D) space inside the W-space sphere, it can interact with 3D obstacles that may appear inside the workspace sphere, thus presenting a potential collision avoidance issue. The fact that one end of link l 2 is fixed (at the base J 1 ) and the motion of its other end is limited to the sphere surface constrains the link interaction with obstacles significantly. In terms of motion planning, this is equivalent to motion along a curve rather than around a “real” 3D object. This means that the 2D motion planning algorithms of Section 5.2 fully apply here. Proceeding in this direction, we want to chose an M-line, the line that the arm endpoint would go through if no obstacle interfered with the arm motion. Since the 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 positive direction of change of angle θ 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: θ 1 θ 2 M 1 : ++ M 2 : −+ M 3 : +− M 4 : −− The choice of the M-line and the motion planning procedure will proceed accord- ing to the RR-Arm Algorithm (Section 5.2.2). 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 in Figure 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, l 1 . Links l 1 and l 2 lie in the same plane—in general, depending on link l 1 shape, in parallel planes. The arm’s endpoint moves along the surface of a torus, and so its W -space is a torus. ∗ This torus may or may not have a hole depending on the relation between the lengths of links l 1 and l 2 : l 2 >l 1 produces a W-space torus with no hole; l 2 <l 1 produces a W-space torus with a hole. Projections of W -space onto the horizontal (xy) and vertical (xz or yz) planes for both cases, l 2 >l 1 and l 2 <l 1 , are shown in Figures 5.21b and 5.21c, respec- tively. ∗ 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 l 2 J 2 J 1 q 2 q 1 q 2 q 1 a z y S z T z T x X S x S y T y S M 1 M 2 q 1 − q 1 + q 2 + q 2 − M 4 M 3 T (a) ( 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.) 224 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS (a) z S l 2 l 1 q 2 q 2 q 1 q 1 a J 1 x y T (b) y x z x (y) l 2 − l 1 l 1 + l 2 l 2 > l 1 l 2 l 1 l 1 (c) x (y) l 1 + l 2 l 1 l 2 l 1 l 2 < l 1 y x z Figure 5.21 The RR arm of Figure 5.19c. (a) The arm design. (b) Two projections of the arm W-space in case l 2 >l 1 . (c) Two projections of the arm W -space in case l 2 <l 1 . RR Arm of Figure 5.19d. Details of this arm configuration are shown in Figure 5.22a. The difference between this configuration and the one in Figure 5.21c is that here link l 2 rotates in the plane perpendicular, rather than par- allel, to link l 1 . In other words, the arm looks like a fan, with link l 2 being the fan propeller’s only blade. This design makes for a somewhat strange workspace: The arm’s endpoint moves along the surface of a truncated sphere of radius  l 2 1 + l 2 2 DISTINCT KINEMATIC CONFIGURATIONS OF RR ARM 225 z P y P y P x y x l 2 l 1 q 2 q 2 q 1 q 1 a z y x D l 2 l 1 l 1 l 2 q 1 q 1 q 2 q 2 (a) (b) Figure 5.22 The RR arm of Figure 5.19d. (a) Arm design. (b) Arm W -space. (Figure 5.22b). Link l 1 moves in the horizontal plane xy.Linkl 2 is always inside the volume D limited by the sphere and a cylinder whose radius is l 1 and whose height is 2l 2 (see Figure 5.22b; the cross section of the volume D is shaded). The body of link l 2 may therefore interact with 3D obstacles that happen to appear in volume D. [...]... , 5 ) and (6 , 10 ), are shown Unlike the RR arm, each point in W -space of our RP arm has only one arm solution That is, there is one-to-one mapping between W -space and the corresponding C-space, as compared to the one-to-two mapping in the case of the RR arm Because of this, and also because virtual lines in W -space are always simple curves, the virtual boundaries in C-space are also simple curves... 5.24b this consists of lines 7-6 -1 2 and 9- 1 0-1 1) 232 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS 3 A segment that is a part of one of the base circles (e.g., line 1 1-1 2, Figure 5.24b); the inside points of this segment cannot be reached by the arm In order to apply the basic path planning procedure to this arm, the algorithm has to reflect specifics of moving along the C-space cylinder Similar to... 12 15 5 14 M1 6 3 A 8 S 2 18,20 S 7 20 18 1 19 C C 21 21 17 q2 = 270° 20 q2 = 180° 18 q=0 19 8 1 0 17,22 4 9 19 6 16 B 10 3,6 S 9 2,7 A 1 8 l1max 10,14 5 4 l1 M2 17 (a) (b) Figure 5.27 PR arm (a) W -space Shown are M-lines M1 and M2 , along with some link positions during the arm’s passing around obstacles A, B, and C (b) The C-space images of the two M-lines, virtual obstacles, and those same positions... complementary M-line is introduced, the numbering of hit and leave points starts over; Lo = S The distance used is a Euclidean distance in W -space Assume the M1 -line is the shorter of the two complementary M-lines The procedure RP-Arm Algorithm includes the following steps 1 Establish an M1 -line as the M-line Set the flag down Set j = 1 Go to Step 2 2 From point Lj −1 , the arm moves along the M-line until... B is limited by the line (6, 7, 13, 9, 10, 11, 12, 6) 230 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS S′ S bS 2 l2max 3 1 10′ 7 10 6′ 4 6 7 + q1 6 9 8 12 4 5 5 q1 = 0 B 3 9 aS A 13 8 O 10 2 3′ 11 T 1 5′ T′ 1' (a) l2 11 l2max T 3 10 S 2 4 B 1 5 6 9 A 8 1′ 7 5′ 6′ 10 ′ q1 = 90 ° q1 = 0 + q1 (b) Figure 5.24 Revolute–prismatic (RP) parallel two-link arm (a) W -space, with obstacles A and B Shadows... intersection with the W -space outer boundary, and a segment of the W -space outer boundary that corresponds to the interval of θ1 complementing that of the M1 -line to 2π This choice for the complementary M-line is largely arbitrary; any M2 -line will do, as long as it is uniquely defined, is computationally simple, and complements the θ1 interval of M1 -line to 2π As the arm moves along the M-line, obstacles... 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 the W -space boundary; in Figure 5.23, this is line 7-1 Of these, the first three segments form a simple open curve, each point of which can be reached by the arm endpoint The fact that the fourth segment... procedure stops 234 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS 5.6 REVOLUTE–PRISMATIC (RP) ARM WITH PERPENDICULAR LINKS The arm is shown in Figure 5.1d An example of the arm’s interaction with two circular obstacles is shown in Figure 5.25a C-space images of the corresponding virtual obstacles and of the M-line are shown in Figure 5.25b The path generated under the motion planning algorithm... point T 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, is introduced, defined as consisting of three parts: two straight-line segments, (S, S ) and (T , T ), continuing the M1 -line segment outwards until... about to develop The Choice of M-Line A straight-line M-line in W -space would be a perfectly legitimate choice for M-line From the practical standpoint, this choice is, however, not convenient because it may make it impossible to maintain continuous motion of both links Try, for example, to follow a straight line between points S and T in Figure 5.27a A discontinuity in the motion of one or both links . obstacle; in Figure 5.23 these are lines 3-2 -1 and 5-6 -7 . • The straight-line segment that is a part of the W -space boundary; in Fig- ure 5.23, this is line 7-1 . Of these, the first three segments. this consists of lines 7-6 -1 2 and 9- 1 0-1 1). 232 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS 3. A segment that is a part of one of the base circles (e.g., line 1 1-1 2, Figure 5.24b); the. truncated sphere. RR Arm of Figure 5.19b. See Figure 5.20a, which shows both revolute joints of this arm, J 1 and J 2 , in the same spot, with the joint axes intersecting at 90 ◦ . Because link l 1 produces