Geometric approach to the accuracy analysis of a class of 3DOF planar parallel robots

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1、Geometric approach to the accuracy analysis of a class of 3-DOF planar parallel robotsAlexander Yu a, Ilian A. Bonev b,*, Paul Zsombor-Murray aa Department of Mechanical Engineering, McGill University, 817 Sherbrooke St.W., Montreal, Canada H3A 2K6bDepartment of Automated Manufacturing Engineering,

2、Ecole de technologie superieure (ETS),1100 Notre-Dame St.W., Montreal, Quebec, Canada H3C 1K3AbstractParallel robots are increasingly being used in industry for precise positioning and alignment. They have the advantage of being rigid, quick, and accurate. With their increasing use comes a need to d

3、evelop a methodology to compare different parallel robot designs. However no simple method exists to adequately compare the accuracy of parallel robots. Certain indices have been used in the past such as dexterity, manipulability and global conditioning index, but none of them works perfectly when a

4、 robot has translational and rotational degrees of freedom. In a direct response to these problems, this paper presents a simple geometric approach to computing the exact local maximum position error and local maximum orientation error, given actuator inaccuracies. This approach works for a class of

5、 three-degree-of-freedom planar fully-parallel robots whose maximal workspace is bounded by circular arcs and line segments and is free of singularities. The approach is illustrated on three particular designs. Keywords: Parallel robots; Dexterity map; Workspace; Positioning; Accuracy; Error analysi

6、s1. IntroductionParallel robots which were once constructed solely in academic laboratories have increasingly been used in industry for positioning and alignment in recent years. With such demand in the market today for these fast and agile machines, new parallel robots are being designed and manufa

7、ctured. However, there are still many key issues regarding the design of new parallel robots, such as optimal design and performance indices. How could one prove that a new parallel robot design is an improvement over existing designs? Is it enough to evaluate a robot based on its workspace? Clearly

8、 in the current industrial climate it is not, as positioning accuracy has become a key issue in many applications.Several well defined performance indices have been developed extensively and applied to serial and parallel robots. However, a recent study 1 reviewed these indices and discussed their s

9、evere inconsistencies when applied to parallel robots with translational and rotational degrees of freedom. The study reviewed the most common indices to optimize parallel robots: the dexterity index 2, the various condition numbers applied to it to increase its accuracy such as the two-norm or the

10、Frobenius number, and the global conditioning index that is computed over the complete workspace of the robot 3. The conclusion of the paper is that these indices should not be used for parallel robots with mixed types of degrees of freedom (translations and rotations).When the authors designed a ne

11、w three-degree-of-freedom (3-DOF) planar parallel robot, they compared it to a similar design using the dexterity index 4. This comparison was somewhat fair, because the designs allow the use of identical dimensions. However should one change the magnitude of the units (e.g., from cm to mm), the num

12、bers within the index would change dramatically. The higher value for the length units would essentially make the dexterity of a robot closer to 0 as shown in Fig. 1.The dexterity indices are very sensitive to both the type of units used and their magnitudes because of the dependence on the Jacobian

13、 matrix. This matrix also mixes non-invariant functions such as translational and rotational capabilities. A possible solution to this problem is the addition of condition numbers, however with each condition number there are particular problems as described in 1.The global conditioning index (GCI)

14、can be used to evaluate a robot over its workspace, which can be used for the optimal design of a robot. However, there remain two problems with this index. Firstly, it is still dependant on a condition number whose problems were outlined in 1. Secondly, it is computationally-intensive.Obviously, th

15、e best accuracy measure for industrial parallel robots would be the local maximum position error and local maximum orientation error, given actuator inaccuracies (input errors), or some generalization of this (e.g., mean value and variance of the errors over a specific workspace). A general method t

16、hat can be used for calculating these errors based on interval analysis was proposed recently in 5. However, this method is computationally-intensive and gives no kinematic insight into the problem of optimal design In contrast, this paper presents a simple geometric approach for computing the exact

17、 local maximum position error and local maximum orientation error for a class of 3-DOF planar fully-parallel robots, whose maximal workspace is bounded by circular arcs and line segments and is free of singularities. The proposed approach is not only faster than any other method (for the particular

18、class of parallel robots) but also brings valuable kinematic insight.The approach is illustrated on three particular designs that are arguable among the best candidates for micro-positioning over a relatively large workspace:1. A new parallel robot, named PreXYT, designed and constructed at Ecole de

19、 technologie superieure (ETS) that has a unique 2-PRP/1-PPR configuration (P and R stand for passive prismatic and revolute joints, respectively, while P stands for an actuated prismatic joint). A CAD model of PreXYT is shown in Fig. 2a.2. Hephaists 3-PRP parallel robot, designed by the Japanese com

20、pany Hephaist Seiko and currently in commercial use. A photo of the industrial model is shown in Fig. 2b.3. Star-Triangle parallel robot, another 3-PRP parallel robot designed at LIRMM in France 6. This robot is a more optimal design of the double-triangular parallel manipulator 7.The remainder of t

21、his paper is organized as follows. The next section presents the proposed geometric approach. Then, Section 3 presents the inverse and direct kinematic equations for all three parallel robots whose accuracy will be studied in this paper. Section 4 briefly describes the geometry of the constant-orien

22、tation workspace for each of the three robots. Finally, Section 5 applies the proposed geometric method for computing the local maximum position and orientation errors. Conclusions are given in the last section.2. Geometric method for computing output errorsConsider a 3-DOF fully-parallel planar rob

23、ot at a desired (nominal) configuration. Let x, y, and denote the nominal position and nominal orientation of the mobile platform and p1, p2, and p3 denote the nominal active-joint variables. Due to actuator inaccuracies of up to s, the actual active-joint variables are somewhere in the ranges p, s,

24、 p, + s (i = 1,2,3). Therefore, the actual position and orientation of the mobile platform are x + Ax, y + Ay, and + A(f, respectively. The question is, given the nominal configuration of the robot (x,y,(f,p1,p2,P3) and the actuator inaccuracy s, how much is the maximum position error, i.e., |).In o

25、rder to compute these errors, we basically need to find the values of the active-joint variables for which these errors occur. The greatest mistake would be to assume that whatever the robot and its nominal configuration, the maximum position error occurs when each of the active-joint variables is s

26、ubjected to a maximum input error, i.e., +s or s. Indeed, in 8, it was proven algebraically that the maximum orientation angle of a 3-DOF planar parallel robot may occur at a Type 1 (serial) or a Type 2 (parallel) singularity, or when two leg wrenches are parallel, for active-joint variables that ar

27、e inside the input error intervals. Similarly, it was pro-ven that not all active-joint variables need to be at the limits of their input error intervals for a maximum position error.Naturally, though there are exceptions, 3-DOF planar parallel robots that are used for precision positioning operate

28、far from Type 1 singularities and certainly far from Type 2 singularities (if such even exist). Furthermore, it is simple to determine whether configurations for which two leg wrenches are parallel correspond to a local maximum of the orientation error and to design the robot in such a way that no s

29、uch configurations exist. Therefore, for such practical 3-DOF planar parallel robot, the local maximum orientation error occurs at one of the eight combinations of active-joint variables with +s or s input errors.Now, finding this local maximum position error is equivalent to finding the point from

30、the uncertainty zone of the platform center that is farthest from the nominal position of the mobile platform. This uncertainty zone is basically the maximal workspace of the robot obtained by sweeping the active-joint variables in the set of intervals p, s, p, + s. Obviously, the point that we are

31、looking for will be on the boundary of the maximal workspace.A geometric algorithm for computing this boundary is presented in 9, but we will not discuss it here in detail. We need only mention that this boundary is composed of segments of curves that correspond to configurations in which at least o

32、ne leg is at a Type 1 singularity (which we exclude from our study) or at an active-joint limit. When these curves are line segments or circular arcs, it will be very simple to find the point that is farthest from the nominal position of the mobile platform. This point will be generally an intersect

33、ion point on the boundary of the maximal workspace.In what follows, three examples will be studied in order to illustrate the proposed geometric approach.3. Inverse and direct kinematic analysisReferring to Fig. 3a-c, a base reference frame Oxy is fixed to the ground and defines a plane of motion fo

34、r each planar parallel robot. Similarly, a mobile reference frame Cxy is fixed to the mobile platform and in the same plane as Oxy. Let At be a point on the axis of the revolute joint of leg i (in this paper, i= 1,2,3) and in the plane of Oxy.Referring to Fig. 3a and b, the base j-axis is chosen alo

35、ng the path of motion of point A2, while the mobile x-axis is chosen along the line A2A3. In Fig. 3a, the origin C coincides with point A1, while in Fig. 3b, the origin C is placed so that point A1 moves along the j-axis. For both robots, s is the distance between the parallel paths of points A2 and

36、 A3, while in the Hephaists alignment stage, h is the distance between the base x-axis and the path of point A1.Referring to Fig. 3c, let points Ot be located at the vertices of an equilateral triangle fixed in the base. Let the origin O of the base frame coincide with 01, and let the base x-axis be

37、 along the line 0102. Let also the origin C be at the intersection of the three concurrent lines in the mobile platform, along which points At move. These three lines make up equal angles. Finally, the mobile y-axis is chosen along the line A1C. Let Pi be the active-joint variables representing dire

38、cted distances, defined as follows. For PreXYT and Hephaists alignment stage (Fig. 3a and b), p1 is the directed distance from the base y-axis to point A1, while p2 and P3 are the directed distances from the base x-axis to points A2 and A3, respectively. Finally, for the Star-Triangle robot (Fig. 3c

39、), pi is the directed distance from Oi to Ai minus a constant positive offset d. Indeed, in the Start-Triangle robot, no mechanical design would allow point Ai to reach point Oi.3.1.PreXYTGiven the active-joint variables, it is straightforward to uniquely define the position and orientation of the m

40、obile platform. The orientation angle is easily obtained as while the position of the mobile platform is given byAs one can observe, the direct kinematics of PreXYT are very simple and partially decoupled.The inverse kinematic analysis is also trivial. Given the position and orientation of the mobil

41、e platform, the active-joint variables are obtained as Obviously, PreXYT has no singularities (provided that s is non-zero).3.2.Hephaists parallel robotGiven the active-joint variables, it is simple to uniquely define the position and orientation of the mobile platform. The equation of the orientati

42、on angle is the same as Eq. (1). The position of the mobile platform is the intersection between line A2A3 and the line passing through point A1 and normal to A2A3. The resulting equations for x and y are thereforeAs one can observe, the direct kinematics of Hephaists parallel robot are more complex

43、 and highly coupled. The inverse kinematics are easier to solve for. Given the position and orientation of the mobile platform, the active-joint variables are obtained asSince Eqs. (7) (11) are always defined (assuming s 0), it is evident that this parallel robot too has no singularities. Note, that

44、 this is quite an advantage over most planar parallel robots, which have singularities.3.3. Star-Triangle parallel robotGiven the active-joint variables, we are able to uniquely define the position and orientation of the mobile platform through this direct kinematic method used in 10. Referring to F

45、ig. 4, the position of Ccan be easily obtained through the following geometric construction based on the notion of the first Fermat pointSince in triangle A1A2A3, none of the angles is greater that 120 (because points At cannot move outside the sides of triangle O1O2O3), equilateral triangles are dr

46、awn outside of it. The outmost vertices of these triangles are denoted as Qt (see Fig. 4). Then lines QtAt make 120 angles and intersect at one point, the so-called first Fermat point. This point is the mobile frames origin C.While there is only solution for the position of the mobile platform, ther

47、e are two possibilities for the orientation angle ( and + 180). Obviously, however, only one of these two solutions is feasible (the one for which 90 90).To find the coordinates of point C and the orientation angle 9, the following simple calculations need to be performed. Let q, denote the vector c

48、onnecting point At to point Q, and a, = xAi, yA is the vector connecting point O to point At. Therefore, it can be easily shown that vector q, can be written as. The orientation of the mobile platform can be found by measuring the angle between line A1C and the base y-axis The inverse kinematics of

49、this device is also easily solved for. Let c = x,y be the vector connecting the base origin O to the mobile frame origin C, b, be the unit length along 0,4, and p, be the unit length along CAt. Then, it can be easily shown that where d is the offset between the vertices of triangle O1O2O3 and the in

50、itial positions of the corresponding linear actuators (see Fig. 3c). 4.Constant-orientation workspace analysisThere exists a simple geometric method for computing the constant-orientation (position) workspace of planar parallel robots 9. Although, calculating the constant-orientation workspace for t

51、he three robots is not necessary for computing the accuracy of these robots, this workspace analysis shows that there is no simple relationship between accuracy and workspace shape and dimensions.To allow for fair comparison, it will be assumed that the only limits restraining the workspace of the r

52、obots are the actuator limits. Under these conditions, the constant-orientation workspaces for all three planar parallel robots can be easily obtained geometrically, as shown in Fig. 5, where the constant-orientation workspaces for a given orientation are shown as the hatched regions.From here, it i

53、s obvious that the constant-orientation workspace of PreXYT is greater than the constant-orientation workspace of Hephaists robot, for any non-zero orientation. Furthermore, the constant-orientation workspace of PreXYT is always centered between actuators 2 and 3, while the location of the constant-

54、orientation workspace of Hephaists robot is hugely varying. This means that the region in which each point is accessible with at least one orientation in a given range, E max, maxl, is much larger for PreXYT. However, the so-called maximal workspace (the set of all attainable positions) is much grea

55、ter for the Hephaists parallel robot.Unfortunately, Fig. 5c does not show clearly whether the constant-orientation workspace of the Star-Triangle parallel robot is larger than that for the other two robots, because the orientation is not the same, nor are the limits of the actuators. In fact, for th

56、e zero orientation, the constant-orientation workspace of PreXYT is larger than that of the Star-Triangle parallel robot, but beyond some orientation angle, the situation is reverse. However, it is clear that the maximal workspace of the Star-Triangle parallel robot is greater than that of PreXYT (w

57、hich is a square whose side is the actuators travel).Intuitively, the larger is the maximal workspace of a robot, the smaller is its positioning accuracy. But if only maximum position errors are considered (as done in this paper), then it is not the area of the maximal workspace that defines accurac

58、y but its extreme dimensions, as it will be seen in the next section. 5. Error analysisFor comparison reasons, it will be considered that all three parallel robots use the same actuators, whose stroke is 500 mm. Let s = 0.05 mm be the accuracy of these actuators, meaning that if the nominal value of

59、 active-variable i is pi, then its real value is somewhere in the range /, ,p, + s. Let also the offset d for the Star-Triangle parallel robot be 100 mm. Based on these assumptions, it will now be shown that the maximum orientation and position errors can be found analytically using a simple geometr

60、ic method. 5.1. PreXYT5.1.1. Maximum orientation errorReferring to Fig. 6a, it is obvious that the maximum orientation error does not depend on the nominal position of the mobile platform and occurs when actuators 2 and 3 are at p2 s and p3 =f e, respectively (depending on the sign of the nominal or

61、ientation angle, it will be one of the two combinations). To solve analytically for the orientation error, a simple system of similar right triangles should be analyzed. Skipping the details, it can be shown that the maximum orientation angle is where is the nominal orientation angle. Fig. 7 shows a

62、 plot of the maximum orientation error of PreXYT as a function of nominal orientation. Obviously, it shows clearly that when increases, rmax decreases.5.1.2. Maximum position errorReferring again to Fig. 6a, the hatched area is the maximal workspace for the considered ranges of the active-joint vari

63、ables: p, s,p, + s. Indeed, this is the region where point C can be located. Its geometric computation 9 is fairly simple for 3-DOF parallel robots whose legs have two prismatic joints and one rev-olute joint. The boundaries of the maximal workspace are the curves described by point C, when two actu

64、ators are at a limit. Note that for a general 3-DOF planar parallel robot, the boundaries of the maximal workspace may also include the curves for which a leg is at a singularity, which complicates the computation.Fortunately, however, none of the legs of all three parallel robots considered in this

65、 paper can be singular (without disassembling the robot).To find the maximum position error, the maximum distance between the nominal position and the boundary of the maximum workspace should be found. This would be very difficult to do analytically for a general 3-DOF planar parallel robot, where the boundaries of the maximal workspace could be segments of ellipses or even sextics. However, for 3-DOF planar parallel robots whose legs hav