IROS2019国际学术会议论文集2402

Kinematically Redundant Hybrid Robots with Simple Singularity Conditions and Analytical Inverse Kinematic Solutions Kefei Wen and Cl ement Gosselin AbstractThe rotational workspace of a parallel manipula- tor is mainly limited by type II singularities. In this paper, we present an approach to synthesize three-legged planar 3 + n and spatial 6 + n degree-of-freedom kinematically redundant hybrid robots with very simple or vanishing type II singularity conditions. Here n denotes the number of redundancies. Moreover, the inverse kinematic problem of the proposed architectures can be solved analytically. Because of these advantages, such redundant robots are well suited for applications in physical human-robot interaction, where large rotational workspaces are typically needed. I. INTRODUCTION A parallel manipulator generally consists of a moving platform connected to the base through several identical legs. The mapping between the actuated joint velocities () and the Cartesian end-effector velocities (t) can be described as Jt = K(1) where J and K are the Jacobian matrices. The performance of smooth trajectories of the moving platform may be prevented at some special confi gurations, which are mainly classifi ed as the type I (matrix K is rank defi cient) and type II (matrix J is rank defi cient) singularities 1. In most cases, the former type of singularities can be determined intuitively, whereas the latter are more diffi cult to determine and typically occur inside the translational workspace of non-redundant parallel robots, thereby limiting the rotational workspace 2. Parallel robots have the potential to be used as physical human-robot interaction (pHRI) devices. For instance, in 3 a three-degree-of-freedom (3-DoF) mini low-impedance passive mechanism is exploited to effortlessly and intuitively control a high-impedance active macro gantry manipulator. The Jacobian matrix of the mini mechanism 3 is the 33 identity matrix because of the kinematically decoupled structure of the mechanism 4. Thus, it is singularity-free throughout the workspace. Another example of singularity- free non-redundant parallel robot can be found in 5, the type II singularities of the robot are avoided by conducting a specifi c kinematic and geometric design of the architecture. However, applications of most other non-redundant parallel mechanisms are still limited mainly due to the inevitable type II singularities. *The fi nancial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and of the Canada Research Chair program is gratefully achnowledged. The authors are with the Department of Mechanical Engineering, Uni- versit e Laval, Qu ebec, QC, Canada,kefei.wen.1ulaval.ca; gosselingmc.ulaval.ca A planar kinematically redundant robot with four PRR legs was proposed in 6. The robot is constructed by replacing one of the three PRR legs of a 3-PRR planar robot by a kinematically redundant one. By taking advantage of this redundancy, the robot is singularity-free throughout the workspace. However, type II singularities are not completely avoided. Indeed, such a singularity may still occur when the two non-redundant legs are aligned, though this confi guration corresponds to the workspace boundary. In 7, it is shown that all type II singularities which may occur in a Gough- Stewart platform are avoidable, if at least three of the six legs are replaced by the proposed kinematically redundant architecture. However, one drawback is that the inverse kinematics (IK) is solvable only using the numerical method 8, which is often the case for kinematically redundant robots. These issues are overcome if a robot is built using the approach proposed here. In this paper, we propose a class of planar and spatial kinematically redundant hybrid robots (KRHRs). In the pro- posed architecture, the platform is connected to the base through three identical kinematically redundant legs. Each leg can be regarded as a combination of a fully actuated sub- leg and a redundant link. We introduce two different forms of redundant links, namely: revolute/spherical-revolute and revolute/spherical-prismatic links1. The links are attached to the moving platform in three different arrangements by revolute or prismatic joints (see Fig. 1). The axes of the revolute joints fi xed on the platform shown in Fig. 1a are orthogonal to the platform plane. The composed KRHRs are completely type II singularity-free, while the IK can be solved analytically. These will be discussed in detail in the following sections. The rest of this paper is organized as follows. In Section II, the velocity equations of the proposed planar KRHRs are developed and the singularity conditions of the robots are investigated. Feasibility of using other types of planar sub- legs to compose the planar KRHRs are explained. Section III focuses on the development of the velocity equations and singularity analysis of the spatial KRHRs with three identical 3-DoF revolute joint actuated serial sub-legs. Different types of mechanisms that can be used as sub-legs for spatial KRHRs are enumerated in Section IV. Methods of solving the inverse kinematic problem of both planar and spatial KRHRs are provided in Section V. Finally, conclusions are drawn. 1Here, revolute/spherical means that in planar robots a revolute joint is used to connect the distal end of the sub-leg to the redundant link while in spatial robots a spherical joint is used for this purpose. IEEE Robotics and Automation Letters (RAL) paper presented at the 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 Copyright 2019 IEEE (a)(b)(c) Fig. 1.Three different assembly arrangements of the platform and the redundant links: (a) R/S-R, (b) R/S-P, (c) R/S-P. p bi x y x ri l ai O i Fig. 2.Kinematic modelling of the 3-RRRR planar KRHR. II. PLANAR ARCHITECTURES A. Velocity Equations of Architectures with Three Identical RR Sub-Legs We start the discussion with a 3-RRRR planar KRHR shown in Fig. 2. Here R and R denote the actuated and passive revolute joints. The length of the redundant RR link in each leg can be obtained as (p + Qbi ri)T(p + Qbi ri) = l2,i = 1,2,3(2) where Q is the 22 orientation matrix of the platform frame with respect to the base frame, p is the position vector of the origin of the platform frame and riis the position vector of the third revolute joint of the ith leg. The time derivative of (2) yields aT i( p + Qbi ri) = 0(3) where ai= p + Qbi ri(4) and where the product aT i r ican be written as aT i r i= aTiMii, (5) where Miis the 2 2 Jacobian matrix of the ith serial RR sub-leg and i= i1 i2T. Moreover, one has Q = EQ(6) where is the angular velocity of the platform and where E = ?0 1 10 ? .(7) p x ri x y O Fig. 3.Kinematic modelling of the 3-RRRP planar KRHR with platform assembly arrangement (b) of Figure 1. p ri x y O ni di Fig. 4.Kinematic modelling of the 3-RRRP planar KRHR with platform assembly arrangement (c) of Figure 1. Then, defi ning ci= aT i EQbiand then substituting (5) and (6) into (3) yields the velocity equations of the robot, namely Jt = K. In which the Jacobian matrices are respectively J = aT 1 c1 aT 2 c2 aT 3 c3 , K = aT 1M1 00 0aT 2M2 0 00aT 3M3 , (8) where 0 is a 12 zero matrix. Matrix J is of dimension 33 while matrix K is of dimension 36. And t = pT, Tand = T 1 T 2 T 3T. In addition, each row in the Jacobian J represents a planar Pl ucker line along the direction of vector aiand passes through both the passive revolute joints in the same leg. For the assembly arrangements of the planar 3-RRRP architectures (see Figs. 3 and 4), the axis of the ith prismatic (abbreviated as P) joint is always perpendicular to a unit vector niwhich is expressed in the platform frame. This can be written as dT i(Qni) = 0. (9) Vector diin the platform assembly arrangement (b) can be written as di= ri+ ai p Qbi(10) in which ai= lQniand biis a constant vector pointing from the origin of the platform frame to an arbitrary point on the axis of the ith prismatic joint, expressed in the platform (a)(b)(c) Fig. 5.Three Pl ucker lines in each platform assembly arrangement. frame. Differentiating (9) with respect to time and conducting a derivation similar to the one presented above for the 3- RRRR architecture, one can obtain the following Jacobian matrices J = nT 1QT c1 nT 2QT c2 nT 3QT c3 (11) K = nT 1QTM1 00 0nT 2QTM2 0 00nT 3QTM3 (12) in which the scalars ci,i = 1,2,3 are ci= nT iQ T(EQb i E Tdi). (13) It can be easily observed that each Pl ucker line in J passes through the passive revolute and prismatic joints and is perpendicular to the prismatic joint axis in the associated leg. For the platform assembly arrangement (c), the vector diis written as di= ri p.(14) In this case the Jacobian matrices J and K have the same forms as (11) and (12) except that ci,i = 1,2,3 become ci= nT iQ TETdi (15) and the Pl ucker lines in J have the same properties as those in assembly arrangement (b). B. Singularity Analysis of Architectures with Three Identical RR Sub-Legs The Pl ucker lines for each platform assembly arrangement are illustrated in Fig. 5 by dashed lines. A type II singularity occurs whenever the three Pl ucker lines intersect at one common point (may happen in assembly arrangements (a) and (b) or are parallel to each other (may happen only in assembly arrangement (a). Such singularities for assembly arrangement (c) are automatically avoided due to mechanical limits. For the other cases, type II singularities are also readily avoided. Since the sub-legs are fully actuated, it is possible to control the geometric relationship between each redundant link and the moving platform independently during motions, to ensure that the robot is never in a type II singular confi guration. For instance, a simple approach is to predetermine a non-singular geometric relationship between the redundant links and the moving platform and maintain it unchanged. Moreover, if the legs of the proposed planar KRHR are stacked in different planes parallel to x y z si z y x bi Ri1 Ri2 Ri3 Si Ri4 Q p i ai li1 li2 li3 li4 xi1 xi2 xi3 i1 i2 i3 Fig. 6.Kinematic modelling of one leg of the 3-RRRSR spatial KRHR. each other, theoretically, the rotational workspace could be unlimited, whereas the rotational workspace of the 3-RRR non-redundant counterpart would mostly still be limited by type II singularities, which can be visualized by determining the type II singularity loci 9. Additionally, the type I singularities, corresponding to det(Mi) = 0, can be easily determined since they occur simply when the ith RR sub-leg is fully expanded or folded. C. Other Feasible Planar Sub-Legs A feasible sub-leg could be a serial, parallel, or redundant planar robot which has at least two translational DoFs. For each platform assembly arrangement, using different forms of sub-legs only changes the Jacobian matrix Mi. Thereby, type II singularity conditions are preserved. Most planar mechanisms have very simple structures, thus we omit the kinematic analysis in this paper. Special attention should be paid to the planar parallel mechanisms because they may encounter both type I and type II singularities. However, the singular confi gurations can be revealed easily via screw theory 10 or Grassmann-Cayley algebra 11. III. SPATIAL ARCHITECTURES WITH RRR SUB-LEGS A. Velocity Equations In each leg of the spatial 3-RRRSR KRHR (see Fig. 6), the axes of joints Ri2and Ri3,(i = 1,2,3) are parallel to each other and perpendicular to the axis of joint Ri1. Parameters lij(j = 1,2,3,4) denote the length of the jth link in the ith leg and Sirepresents the passive spherical joint of leg i. The derivation for the velocity equations of this archi- tecture has been presented in 12 and is briefl y recalled here. For each leg, a constraint equation on the length of the redundant link can be written as (p + Qbi si)T(p + Qbi si) = l2 i4 (16) where siis the position vector of the centre of the ith spher- ical joint. Also, the axis of joint Ri4is always perpendicular to the direction of the ith redundant link, thus qT z (p + Qbi si) = 0(17) where the unit vector qzis qz= Qz0O0(18) where z0O0= 0 0 1Trepresents the z0axis expressed in the platform frame and is perpendicular to the platform plane. The Cartesian velocities of the platform and actuated joint velocities can be respectively expressed as t = pTTT(19) = T 1 T 2 T 3 T (20) with i= i1 i2 i3T(see Fig. 6). The dimension of vector t is 61 while it is 91 for vector . The Jacobian matrices can be obtained by differentiating the constraint equations with respect to time, which yields J = aT 1 (Qb1) a1T qT z (u1 qz)T aT 2 (Qb2) a2T qT z (u2 qz)T aT 3 (Qb3) a3T qT z (u3 qz)T (21) which is of dimension 66 and in which ai= p + Qbi si(22) ui= Qbi ai(23) and K = aT 1M1 00 qT zM1 00 0aT 2M2 0 0qT zM2 0 00aT 3M3 00qT zM3 (24) which is of dimension 69 while 0 is the 13 zero matrix. Also Mi= ei1 ri1ei2 ri2ei3 ri3(25) is the Jacobian matrix of the RRR chain of the ith leg, in which the unit vector eik,(k = 1,2,3) represents the direction of the axis of the kth actuated joint and rikis a position vector pointing from the centre of this actuated joint to the centre of joint Si. All these quantities are expressed in the base frame. The derivations of the velocity equations for the 3-RRRSP KRHRs with both platform assembly arrangements (b) and (c) (see Figs. 7 and 8) are simpler than that of the 3-RRRSR architecture (although the detailed derivation is not provided here). For the architecture schematically shown in Fig. 7, it can be observed that a vector along the axis of the ith passive prismatic joint is always perpendicular to a plane formed by two mutually perpendicular vectors Qniand qz, which can be written mathematically as dT i(Qni) = 0 (26) dT iqz= 0 (27) x y z si Ri1 Ri2 Ri3 Si li2 li3 li4 Q p z y x bi ni ai Fig. 7.Kinematic modelling of one leg of the 3-RRRSP spatial KRHR with platform assembly arrangement (b). x y z si Ri1 Ri2 Ri3 Si Q p ni Fig. 8.Kinematic modelling of one leg of the 3-RRRSP spatial KRHR with platform assembly arrangement (c). where di= si+ ai p Qbi(28) with ai= li4Qni. Also, biis a constant vector directed from the origin of the platform frame to an arbitrary point on the axis of ith prismatic joint, niis a constant unit vector on the platform plane which is perpendicular to the ith prismatic joint and expressed in the platform frame, and qzhas been defi ned in (18). If vector biis chosen to be parallel to vector ni , (26) can be further simplifi ed to (si p)T(Qni) = b li4(29) where the constant scalars b and li4are obtained from b = (Qbi)T(Qni),li4= aT i(Qni). (30) On the other hand, since vector qzis always perpendicular to vectors aiand Qbi, (27) is the same as (si p)Tqz= 0.(31) The time derivative of (29) gives (Qni)T p (si p)T(Qni) = (Qni)T s i. (32) Since (si p)T(Qni) = (si p)T(Qni) = (si p)T (Qni) = (Qni) (si p)T(33) and si= Mii, (32) can be rewritten as (Qni)T p + (si p) (Qni)T = (Qni)TMii. (34) Similarly, the time derivative of (31) yields qT z p (si p)T qz= qT z s i, (35) here (sip)T qz= (sip)T(qz) = qz(sip)T, (36) thus (35) can be rewritten as qT z p + (si p) qzT = qT zMi i.(37) The Jacobian matrices can be obtained by combining the corresponding terms in (34) and (37), yielding J = nT 1QT (s1 p) (Qn1)T qT z (s1 p) qzT nT 2QT (s2 p) (Qn2)T qT z (s2 p) qzT nT 3QT (s3 p) (Qn3)T qT z (s3 p) qzT (38) and K = nT 1QTM1 00 qT zM1 00 0nT 2QTM2 0 0qT zM2 0 00nT 3QTM3 00qT zM3 .(39) The constraint conditions of the architecture illustrated in Fig. 8 can be described similarly to those of the above dis- cussed 3-RRRSP KRHR. Thus, one can write the constraint equations as follows (si p)T(Qni) = 0(40) (si p)Tqz= 0.(41) Clearly, one obtains the Jacobian matrices J and K with the same forms as (38) and (39) after differentiating the above equations with respect to time. B. Singularity Analysis Spatial KRHRs may suffer from the shoulder singularity, occurring when the centre of joint Silies on the axis of joint Ri1, and elbow singularity, occurring when the last two links in a RRR leg are fully extended or folded. In such confi gurations, det(Mi) = 0, which also refers to the type I singularities. The robots translational workspace is more likely to be limited by such singularities. Thus, in a practical design, it would be advisable to tilt joint Ri1so that it makes an obtuse angle () with the base (see Fig. 8). A reasonable (a)(b)(c) Fig. 9.Six Pl ucker lines in each platform assembly arrangement. range for is 2 : the larger the angle, the larger the singularity-free translational workspace 12. The complexity of type II singularity conditions is sig- nifi cantly reduced thanks to the types of redundant links as well as the platform assembly arrangements. Each row of Jacobian J represents a spatial Pl ucker line. For a spherical- revolute redundant link, one Pl ucker line lies on the platform plane and passes through both joints, the other one is parallel to vector qz. For a spherical-prismatic redundant link, one line passes through both joints and is perpendicular to the prismatic joint and the other one is parallel to vector qz. In each case, two lines intersect at the centre of the spherical joint. As described in Fig. 9, the dashed lines represent the coplanar Pl ucker lines; while the spatial parallel Pl ucker lines (orthogonal to the paper) are indicated by small solid circles. These two groups of lines belong to two orthogonal vector spa