基于幾何方法精確分析一類三自由度的平面并聯(lián)機器人外文文獻翻譯、中英文翻譯

基于幾何方法精確分析一類三自由度的平面并聯(lián)機器人外文文獻翻譯、中英文翻譯,基于,幾何,方法,法子,精確,分析,一類,自由度,平面,并聯(lián),機器人,外文,文獻,翻譯,中英文 Received 12 June 2006; received in revised form 11 December 2006; accepted 16 March 2007 Keywords: Parallel robots; Dexterity map; Workspace; Positioning; Accuracy; Error analysis has become a key issue in many applications. * Corresponding author. E-mail address: ilian.bonev@etsmtl.ca (I.A. Bonev). Available online at Mechanism and Machine Theory 43 (2008) 364–375 Mechanism and Machine Theory 0094-114X/$ - see front matter C211 2007 Elsevier Ltd. All rights reserved. 1. Introduction Parallel 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 manufactured. 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 eval- uate a robot based on its workspace? Clearly in the current industrial climate it is not, as positioning accuracy Available online 3 May 2007 Abstract Parallel 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 develop a methodology to compare di?erent 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 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 three-degree-of-freedom planar fully-par- allel 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. C211 2007 Elsevier Ltd. All rights reserved. Geometric approach to the accuracy analysis of a class of 3-DOF planar parallel robots Alexander Yu a , Ilian A. Bonev b, * , Paul Zsombor-Murray a a Department of Mechanical Engineering, McGill University, 817 Sherbrooke St.W., Montreal, Canada H3A 2K6 b Department of Automated Manufacturing Engineering, E ′ cole de technologie supe′rieure (E ′ TS), 1100 Notre-Dame St.W., Montreal, Quebec, Canada H3C 1K3 doi:10.1016/j.mechmachtheory.2007.03.002 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 severe 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 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 new 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 numbers within the index would change dramatically. The higher value for the length units would essen- tially 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 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) 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 depen- dant on a condition number whose problems were outlined in [1]. Secondly, it is computationally-intensive. Obviously, the 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 that 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. 0. 0038 0. 00380. 0. 004 0. 0042 0. 0042 0. 0044 0 4 4 0. 0046 4 6 0. 0048 0. 0048 0. 0049 0. 0049 1 500 0 . 0 4 1 0 . 0. 00 49 3 500 ab A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 365 0. 0038 0. 0038 0. 0038 0. 0038 004 0. 004 0. 004 0. 004 0 . 0 0 4 0. 0042 0 . 0 0 4 2 0. 0042 0. 0042 0. 0044 0. 0044 0 . 0 0. 0044 0. 0044 0. 0046 0. 0046 0 .0 0 0. 0046 0. 0046 0. 0048 0. 0048 0. 0048 0. 0049 0. 0049 0 . 0 0 4 9 0 . 0 0 4 9 0. 004941 0. 004941 0 . 0 0 4 9 4 0. 004941 x (mm) y( m m ) 0 100 200 300 400 500 600 700 0 100 200 300 400 0 .00 2 5 0 .0 0 2 50 .0 0 27 0 0 027 0 .0 0 2 7 0 . 0 0 2 9 0. 00 29 0. 00 29 0 . 0 0 2 9 0 . 0 0 3 1 0. 00 31 0. 00 31 0 . 0 0 3 1 0 . 0 0 3 3 0. 00 33 0. 00 33 0 . 0 0 3 3 0 . 0 0 3 5 0 .0 0 3 5 0.0 03 5 0 . 0 0 3 5 0. 003 7 0 . 0 0 3 7 0.0037 0 . 0 0 3 7 0 . 00 37 0 3 9 0 . 0 0 3 9 0.0039 0 . 0 0 3 9 0 . 0 0 4 1 0. 00 41 0 .0 0 4 1 0 . 0 0 0 .0 0 4 3 0. 00 43 0 . 0 0 4 3 0 0 4 5 0.0045 0 . 0 0 4 5 0 .0 0 4 7 0 .0 0 4 7 0. 00 49 x (mm) y( m m ) 0 100 200 300 400 500 600 700 0 100 200 300 400 0 . 0 0 5 0.0 05 0 . 0 0 5 0. 0 05 5 0. 00 55 0 . 0 0 5 5 0.0 05 5 0 . 0 0 5 5 0 . 0 0 6 0 .0 0 6 0 .0 0 6 0 .00 6 0 .0 0 6 5 0 . 0 0 6 5 0. 0065 0 .0 0 7 0 . 0 0 7 0 .0 0 7 2 50 . 0 0 7 4 x (mm) y( m m ) 0 100 200 300 400 500 600 700 0 100 200 300 400 500 c Fig. 1. Example of dexterity maps for (a) PreXYT, (b) Hephaist’s parallel robot and (c) the Star-Triangle parallel robot. 366 A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 In contrast, this paper presents a simple geometric approach for computing the exact local maximum posi- tion error and local maximum orientation error for a class of 3-DOF planar fully-parallel robots, whose max- imal 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 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 E ′ cole de technologie supe′rieure (E ′ TS) 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. Hephaist’s 3-PRP parallel robot, designed by the Japanese company Hephaist Seiko and currently in com- mercial 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 this 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-orienta- tion 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. Fig. 2. (a) PreXYT (patent pending) and (b) Hephaist’s NAF3 alignment stage (courtesy of Hephaist Seiko Co., Ltd). 2. Geometric method for computing output errors Consider a 3-DOF fully-parallel planar robot at a desired (nominal) configuration. Let x, y, and / denote the nominal position and nominal orientation of the mobile platform and q 1 , q 2 , and q 3 denote the nominal active-joint variables. Due to actuator inaccuracies of up to ±e, the actual active-joint variables are somewhere in the ranges ?q i C0 e; q i t eC138 (i ? 1; 2;3). Therefore, the actual position and orientation of the mobile platform are x tDx, y tDy, and / tD/, respectively. The question is, given the nominal configuration of the robot ex; y; /; q 1 ; q 2 ; q 3 T and the actuator inaccuracy ±e, how much is the maximum position error, i.e., d max ? maxe ????????????????????? Dx 2 tDy 2 p T, and the maximum orientation error, i.e., r max ? maxejD/jT. In order 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 config- uration, the maximum position error occurs when each of the active-joint variables is subjected to a maximum input error, i.e., +e or C0e. 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 are 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 posi- tion error. Naturally, though there are exceptions, 3-DOF planar parallel robots that are used for precision position- ing operate far from Type 1 singularities and certainly far from Type 2 singularities (if such even exist). Fur- thermore, 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 such 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 +e or C0e input errors. Now, finding this local maximum position error is equivalent to finding the point from 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 ?q i C0 e; q i t eC138. Obviously, the point that we are 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 con- A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 367 figurations in which at least one 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 intersection 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 analysis Referring to Fig. 3a–c, a base reference frame Oxy is fixed to the ground and defines a plane of motion for each planar parallel robot. Similarly, a mobile reference frame Cx 0 y 0 is fixed to the mobile platform and in the same plane as Oxy. Let A i 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 y-axis is chosen along the path of motion of point A 2 , while the mobile x 0 -axis is chosen along the line A 2 A 3 .InFig. 3a, the origin C coincides with point A 1 , while in Fig. 3b, the origin C is placed so that point A 1 moves along the y 0 -axis. For both robots, s is the distance between the par- allel paths of points A 2 and A 3 , while in the Hephaist’s alignment stage, h is the distance between the base x- axis and the path of point A 1 . Referring to Fig. 3c, let points O i be located at the vertices of an equilateral triangle fixed in the base. Let the origin O of the base frame coincide with O 1 , and let the base x-axis be along the line O 1 O 2 . Let also the origin C be at the intersection of the three concurrent lines in the mobile platform, along which points A i move. These three lines make up equal angles. Finally, the mobile y 0 -axis is chosen along the line A 1 C. x y y ’ x ’ s O A 2 A 3 C A 1 C y ’ x ’ x s y A 2 A 1 A 3 O 1 ρ 2 ρ 3 ρ 2 ρ 3 ρ 1 ρ h C y’ x’ x O O 1 y O 2 O 3 A 1 A 2 A 3 2 ρ 3 ρ 1 ρ Fig. 3. Schematics of (a) PreXYT (patent pending), (b) Hephaist’s parallel robot and (c) the Star-Triangle parallel robot. Give mobile As one Give platform 3 2 The the active gularities. Note, that this is quite an advantage over most planar parallel robots, which have singularities. 368 A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 3.3. Star-Triangle parallel robot Given 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 Fig. 4, the position of C can be easily q 1 ? x C0eh C0 yTtan/; e9T q 2 ? y C0 xtan/; e10T q 3 ? y tes C0 xTtan/: e11T Since Eqs. ((7)–(11)) are always defined (assuming s50), it is evident that this parallel robot too has no sin- obtaine inverse kinematics are easier to solve for. Given the position and orientation of the mobile platform, -joint variables are obtained as As one can observe, the direct kinematics of Hephaist’s parallel robot are more complex and highly coupled. s 2 teq 3 C0 q 2 T 2 y ? s 2 q 2 t heq 3 C0 q 2 T 2 t sq 1 eq 3 C0 q 2 T s 2 teq C0 q T 2 : e8T is the intersection between line A 2 A 3 and the line passing through point A 1 and normal to A 2 A 3 . The resulting equations for x and y are therefore x ? seq 1 s teh C0 q 2 Teq 3 C0 q 2 TT ; e7T ’s parallel robot n the active-joint variables, it is simple to uniquely define the position and orientation of the mobile . The equation of the orientation angle is the same as Eq. (1). The position of the mobile platform The inverse kinematic analysis is also trivial. Given the position and orientation of the mobile platform, the active-joint variables are obtained as q 1 ? x; e4T q 2 ? y C0 xtan/; e5T q 3 ? y tes C0 xTtan/: e6T Obviously, PreXYT has no singularities (provided that s is non-zero). 3.2. Hephaist x ? q 1 ; e2T y ? q 2 t q 1 q 3 C0 q 2 s C16C17 : e3T can observe, the direct kinematics of PreXYT are very simple and partially decoupled. / ? tan C01 q 3 C0 q 2 s ; e1T while the position of the mobile platform is given by n the active-joint variables, it is straightforward to uniquely define the position and orientation of the platform. The orientation angle is easily obtained as C16C17 Let q i be the active-joint variables representing directed distances, defined as follows. For PreXYT and Hephaist’s alignment stage (Fig. 3a and b), q 1 is the directed distance from the base y-axis to point A 1 , while q 2 and q 3 are the directed distances from the base x-axis to points A 2 and A 3 , respectively. Finally, for the Star- Triangle robot (Fig. 3c), q i is the directed distance from O i to A i minus a constant positive o?set d. Indeed, in the Start-Triangle robot, no mechanical design would allow point A i to reach point O i . 3.1. PreXYT d through the following geometric construction based on the notion of the first Fermat point. Since sides are denoted which To perfor "# ???p A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 369 where E ? C010 01 C20C21 ; e13T and ei; j; kT?e1; 2; 3T or (2, 3, 1) or (3, 1, 2). Now, taking lines A 1 Q 1 and A 2 Q 2 for example, their intersection point is C and has the following coordinates: x ? ex A 1 y Q 1 C0 x Q 1 y A 1 Tex A 2 C0 x Q 2 TC0ex A 2 y Q 2 C0 x Q 2 y A 2 Tex A 1 C0 x Q 1 T ex A 1 C0 x Q 1 Tey A 2 C0 y Q 2 TC0ex A 2 C0 x Q 2 Tey A 1 C0 y Q 1 T ; e14T ex A 1 y Q 1 C0 x Q 1 y A 1 Tey A 2 C0 y Q 2 TC0ex A 2 y Q 2 C0 x Q 2 y A 2 Tey A 1 C0 y Q 1 T The orient axis The inverse origin Then, where ear a q i C17 x Q i y Q i ? 1 2 ea j t a k Tt 3 2 Eea j C0 a k T; e12T point O to point A i . Therefore, it can be easily shown that vector q i can be written as C090C176 < / <90C176). find the coordinates of point C and the orientation angle h, the following simple calculations need to be med. Let q i denote the vector connecting point A i to point Q i , and a i ??x A i ; y A i C138 T is the vector connecting Fermat point. This point is the mobile frame’s origin C. While there is only solution for the position of the mobile platform, there are two possibilities for the ori- entation angle (/ and / + 180C176). Obviously, however, only one of these two solutions is feasible (the one for in triangle A 1 A 2 A 3 , none of the angles is greater that 120C176 (because points A i cannot move outside the of triangle O 1 O 2 O 3 ), equilateral triangles are drawn outside of it. The outmost vertices of these triangles as Q i (see Fig. 4). Then lines Q i A i make 120C176 angles and intersect at one point, the so-called first 1 O 2 A 1 Q 2 Fig. 4. Solving the direct kinematics of the Star-Triangle parallel robot. x O O 3 C x' y A 3 Q y' A 2 O 3 Q 1 y ? ex A 1 C0 x Q 1 Tey A 2 C0 y Q 2 TC0ex A 2 C0 x Q 2 Tey A 1 C0 y Q 1 T : e15T ation of the mobile platform can be found by measuring the angle between line A 1 C and the base y- / ? a tan 2ey C0 y A 1 ; x C0 x A 1 T: e16T kinematics of this device is also easily solved for. Let c ??x; yC138 T be the vector connecting the base O to the mobile frame origin C, b i be the unit length along O i A i and p i be the unit length along CA i . it can be easily shown that q i ? b T i Eeo i C0 cT b T i Ep i C0 d; e17T d is the o?set between the vertices of triangle O 1 O 2 O 3 and the initial positions of the corresponding lin- ctuators (see Fig. 3c). 4. Constant-orientation workspace analysis There exists a simple geometric method for computing the constant-orientation (position) workspace of planar parallel robots [9]. Although, calculating the constant-orientation workspace for the three robots is not necessary for computing the accuracy of these robots, this workspace analysis shows that there is no sim- ple 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 robots are the actuator limits. Under these conditions, the constant-orientation workspaces for all three planar par- allel robots can be easily obtained geometrically, as shown in Fig. 5, where the constant-orientation work- spaces for a given orientation are shown as the hatched regions. From here, it is obvious that the constant-orientation workspace of PreXYT is greater than the constant- orientation workspace of Hephaist’s robot, for any non-zero orientation. Furthermore, the constant- orientatio