Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 291-300 © 2013 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2012.727 Original Scientific Paper Received for review: 2012-08-06 Received revised form: 2012-10-11 Accepted for publication: 2013-02-14 Analysing Kinematics of a Novel 3CPS Parallel Manipulator Based on Rodrigues Parameters Gang Cheng1* - Peng Xu1 - De-hua Yang2 - Hui Li2 - Hou-guang Liu1 1 China University of Mining and Technology, College of Mechanical and Electrical Engineering, China 2 Chinese Academy of Sciences, National Astronomical Observatories, Nanjing Institute of Astronomical Optics and Technology, China In order to adjust the poses of the segment mirrors and give the correct surface shape to a large aperture telescope, an active adjusting platform for segment mirror with a novel 3CPS parallel manipulator as core module is proposed. The platform has 6 degree-of-freedoms (DOFs) including three translational freedoms and three rotational freedoms. Its kinematics are analysed systematically. By means of the Rodrigues parameters method, the formulae for solving the inverse/forward displacement, the inverse/forward velocity, and the inverse/ forward acceleration kinematics are derived. A numerical simulation of the kinematics model is then carried out combining the topological structure characteristics of the manipulator. The correctness of the kinematics model is verified by an experiment in which the pose of moving platform is measured using a photogrammetric method. Keywords: active adjusting platform, 3CPS parallel manipulator, Rodrigues parameters, kinematics, photogrammetry 0 INTRODUCTION Astronomy seeks the detection of more distant and dim celestial bodies. Large aperture optical systems are significant in astronomy research for their increased light gathering capability and angular resolution in the object space [1]. Therefore, the development of telescopes with large apertures and high imaging quality is necessary. However, as the size of the telescopes increases, they become increasingly sensitive to external disturbances such as thermal gradients, gravity, and wind, and also to internal disturbances from support equipment such as pumps, cryocoolers, and fans [2]. In order to decrease these influences, most large aperture telescopes in the future will be segmented [3], such as the US/Japanese Thirty Meter Telescope and the European-ELT 42m telescope projects. For these telescopes, the large aperture primary mirrors are spliced by many small aperture and thin segment mirrors. The advantages of this approach, where the mass and size of the primary mirror are no longer the main factors affecting the observation results, are obvious. However, comparing the telescope with one entire primary mirror, the main problems with this approach are the position and orientation disorders of the segment mirrors. The development of an active adjusting mechanism for a large number of mirrors with multiple degrees of freedom is urgent. In recent years, parallel manipulators have seen growing applications in robotics, machine tools, positioning systems, measurement devices, and so on [4] to [8]. The classic 6 DOFs parallel manipulator (Stewart platform) has many characteristics, such as simple structure, concise principle, flexible function, high accuracy, and high stiffness. It has also been used in the field of astronomical telescope and instruments [9] to [11]. However, the six driving legs of the Stewart platform have strong coupling movements and the distance between the moving platform and fixed base is usually large. These shortcomings, as well as the structure type of the Stewart platform, are not suitable for making large-scale active adjustments in compact spaces. In order to overcome the defects mentioned above, an active adjusting platform prototype of a segment mirror with a novel 3 CPS parallel manipulator as the core module is proposed. The parallel manipulator is composed of a moving platform, a fixed base and three CPS chains, where the notation CPS denotes the kinematic chain made up of a cylindrical joint, a prismatic joint, and a spherical joint in series. Kinematic analysis is a common basis of dynamic analysis and control system design. The kinematics of parallel manipulators includes inverse kinematics and forward kinematics. Numerous researchers have made contributions to this. Cheng et al. [12] studied the inverse/forward displacement, velocity, and acceleration kinematics of a 3SPS+1PS bionic parallel test platform by means of the unit quaternion method. Lu et al. [13] studied kinematics, statics, and workspaces of a 3R1T 4 DOFs and a 1R3T 4 DOFs parallel manipulators, comprehensively. Gallardo et al. [14] studied the kinematics of modular spatial hyper-redundant manipulators formed from RPS-type limbs based on screw theory and recursive method. Cui et al. [15] analysed the kinematics of a TAU parallel manipulator based on a D-H model and solved *Corr. Author's Address: China University of Mining and Technology, College of Mechanical and Electrical Engineering, 221008, Xuzhou, China, chg@cumt.edu.cn 291 its forward kinematics in closed forms by a Jacobian approximation method. Varedi et al. [16] analysed the kinematics of an offset 3UPU translational parallel manipulator by the homotopy continuation method, which alleviates the drawbacks of traditional numerical techniques, namely: the acquirement of good initial guess values, the problem of convergence, and computing time. The main research methods of kinematics can be divided into analytical and numerical ones. For the parallel manipulators whose structures are not complex, the analytical method can solve the kinematics competently and solution procedures can be fully automated. For parallel manipulators with complex structures, the analytical method is always inadequate and the obtained solutions can be too complex to subsequently analyse. The computational accuracy and speed of the numerical method depend on the complexity of mechanisms and algorithms themselves whose flexibility and portability is quite often poor. Due to the simple mechanical structure, the analytical method is adopted to analyse the kinematics in this paper. The rest of the paper is organized as follows: In section 1, the structure of the active adjusting platform prototype is described and the reference systems of the manipulator are established. In section 2, the inverse/ forward displacement, the inverse/forward velocity, and the inverse/forward acceleration kinematics of the manipulator are studied based on the Rodrigues parameters. In section 3, a numerical simulation of the kinematics analysis is conducted, and lastly the numerical results are validated by experiments. 1 DESCRIPTION OF 3CPS PARALLEL MANIPULATOR The active adjusting platform for the segment mirror considered in this paper is shown in Fig. 1a and the topological structure of its core module, a 3CPS parallel manipulator, is shown in Fig. 1b. Taking the 3CPS parallel manipulator as an analysis object, reference systems for kinematic analysis are established. The absolute coordinate system {B} is fixed on B at point O. The 7-axis of {B} passes through point the Z-axis is perpendicular to B pointing to m. The X-axis can be determined by the other two axes following the right-hand-rule (RHR). Three vertical legs with cylindrical joints (Ci, i =1, 2, 3) are installed symmetrically about point O on B. Every mount point is equidistant from point O and the distance denotes as E. Three horizontal legs with prismatic joints (Pi, i = 1, 2, 3) are fixed on the end points of three vertical legs (Di, i = 1, 2, 3). They rotate around Z-axis in the same direction and the angles between the horizontal legs and the corresponding sides of triangular B are equal. Three equal-length short legs are connected to the horizontal legs and the lengths were unchanged. In the following analysis, the lengths of the short legs can add to the lengths of the vertical legs. The short legs are connected to m with sphere joints (Si, i = 1, 2, 3). The distances of mount points in m are equal to each other. The relative coordinate system {m} is attached to m at point o. They-axis of {m} passes through point a2, the z-axis is perpendicular to m pointing upward. The x, y and z axis follow the RHR. Three spherical joints are installed on m and the distance from mount point to point o is denoted as e. When the length of the vertical legs are equal, namely: hi = h0 (i = 1, 2, 3) and the length of the horizontal legs are equal, namely: li = l0 (i = 1, 2, 3), the manipulator is at the equilibrium position. Fig. 1. The 3CPS parallel manipulator; a) a prototype of the active adjusting platform, and b) the topological structure of the active adjusting platform This prototype has partial DOF-decoupling motion characteristics. The rotations around the X-axis, 7-axis and the translations along the Z-axis are driven by the three vertical leg hi (i = 1, 2, 3), while the translations along the X-axis, the 7-axis and the rotations around Z-axis are driven by the three horizontal legs lj (i = 1, 2, 3), respectively. When the manipulator is at the equilibrium position, the moving platform is parallel to the fixed base. Let 8 be the rotational angle around the Z-axis between the moving platform and the fixed base. According to the cosine formula, the formula for solving 8 is expressed as: d = arc cos( E2 + e -10 2Ee (1) Using a Kutzbach Grübler equation [17], the DOF of 3CPS parallel manipulator is calculated as: F = 6 (n - g -1) + £ fi = 6, (2) where, F is the DOF of the manipulator, n is the number of components, g is the number of kinematics pairs, and f is the degree of freedom of the ith kinematics pair. It can be seen clearly that the parallel manipulator has 6 DOFs including three translational freedoms and three rotational freedoms. 2 MANIPULATOR KINEMATICS ANALYSES 2.1 Inverse/Forward Displacement Analysis Before analysing the kinematics of 3CPS manipulator, the coordinates of the points Ai (i = 1, 2, 3) in {B} and the coordinates of the points ai (i = 1, 2, 3) in {m} and {B} must be determined. They are expressed as: xai ' xa¡ II YAi , aam = yai , aB = Ya¡ zai Za¡ o = Y Z„ ai = Da" (3) where, oB is a vector of point o on m in {B}, (Xo, Yo, Zo) are the components of oB, Do is a rotation transformation matrix from {m} to {B} based on Rodrigues parameters and it can be written as follows [18]: Do = xi y¡ xm y m x„ y„ 1+ 02-00 2 (00-0 ) 2 (3 + 0 ) 2 (00 +®3 ) 1 -®12 + 0 -®32 2(00 -®1 ) 2(0301 -®2 ) 2(0302 + 0 ) 1 0 0 +®32 (4) where, Ä02 = 1 + 0l2 + 2 + 02 and (xh Xm, xm yi,ym,yn, zl, zm, zn) are the nine orientation parameters of m in {B}. By rotation transformation, the absolute coordinates aB of the point ai (i = 1, 2, 3) can be derived from Eqs. (3) and (4). The absolute coordinates Df of the point Di (i = 1, 2, 3) can be expressed as follows: DB = ab + Ra B (5) where, R = [0 0 0; 0 0 0; 0 0 1]. According to the formula of distance between two spatial points, the length of the vertical leg h, (i = 1, 2, 3) and the horizontal leg l, (i = 1, 2, 3) can be derived as follows: h =^(db - Ab )t (Db - Ab ), J(a? - D )T (ai - D ). (6) l = The unit vector gi along the vertical legs h, the unit vector Si along the horizontal legs li and the vector ei of line oai can be expressed as: DB - AB aB - DB e, = a - o (7) The unit vector £ is the tangent vector of the horizontal legs l, rotating around vertical legs h, and can be expressed as follow: & = ^ X Ç, (8) When given three Rodrigues parameters (i = 1, 2, 3) and oB, the inverse displacement parameters (hi, li, gi, äi, ei, i = 1, 2, 3) can be solved from Eqs. (6) and (7). When given the lengths of the six input driving legs hi (i = 1, 2, 3) and li (i = 1, 2, 3), from Eq. (6), the position and orientation parameters of moving platform can be obtained by solving a nonlinear kinematic equations system. 2.2 Inverse/forward Velocity and Jacobian Matrix Let V be the general velocity of m at point o and v be the linear velocity, while m is the angular velocity of m at point o and vi is a linear velocity of m at point ai. Let vh and vl be the input velocities of vertical legs and horizontal legs, respectively. This can be expressed as follows: V = v = vl ca1 V2 , m = w2 _V3 _ ca3 v, = v + ®x e,, Vh1 Vl1 vn Vh2 > Vl = Vl 2 . (9) V 2 _Vh3 _ _ Vl3 _ _ V 3 "l" "0" "0" u = D 0 0 0 , Vo = Do l , Wo = Do 0 0 0 l Let uo, vo and wo be the unit vectors of x, y and z axes of the moving platform in {B}, respectively. These expressions can be obtained by rotation transformation from {m} to {B}: (10) The angular velocity m of m at point o can be expressed as follows: m = uom1 + vom2 + wom3. (11) The fflj, a)2 and m3 are the components of m and can be obtained by dot-multiplying Eq. (11) with (vo*wo), (woxwo) and (uo*vo) at both sides, respectively: m, = (Vo X Wo ) m (Uo X Wo ) m 1 (Vo X Wo ) • Uo 2 (Uo X Wo ) • Vo m = (Uo X Vo) m 3 (Uo X Vo ) • wo' (12) According to the geometrical characteristics of the manipulator, the linear velocity vhi along vertical legs h, (i = 1, 2, 3) and the linear velocity vu along horizontal legs l, (i = 1, 2, 3) can be obtained from Eq. (9) and can be expressed as follows: v,. = v, •qi = [ST ( xS)T]V, (13a) vu = v, 5 =[5/ (e, y.8,)T]v. (13b) By combining Eq. (13a) and Eq. (13b), the formulae for solving the inverse/forward velocities can be obtained and expressed as follows: =JV, J= (^1 T " (^2 )T if (e3 X£ )T (l x51 )T (e2 xS2 )T ^3T (e3 xS3 )T where, J is a velocity Jacobian matrix. From Eqs. (8) and (9), the formulae for solving the tangential velocities of the horizontal legs at point a, when rotating around the vertical legs are expressed as: = V =[^T ( )T ]F. (15) From Eqs. (6) and (15), the angular velocities mCi of the vertical legs are derived as shown below: ®a= f, (i = 1, 2, 3). (16) 2.3 Inverse/forward Acceleration and Hessian Matrix First, a skew symmetric matrix is briefly introduced. Suppose two vectors i and v and a skew symmetric matrix S(i) for ¡: Vx Vx " 0 -Vz Vy Vy ,v = Vy ,S(v)= Vz 0 -Vx .Vz _ Vz _ .-Vy Vx 0 H= /dv ,v= Vv , S (/d)= /dz U -idx . (17) Eq. (17) satisfies the following relationships [19]: pxv = S (v)-v =-S (v)- p, S (p)T =-S (v) -S (V)2 + S (»)-S (V)T = 13x3' (18) where, I3x3 is an order 3 unit matrix. Let A be the general acceleration of m at point o, where a and s are the linear and angular acceleration of m at point o, respectively. Let ah and al be the input accelerations of the vertical legs and the horizontal legs, respectively. They can be expressed as follows: v v V A= a = a £ x x a , s = £ y y a z £z _ ah1 ai1 ah = ah2 = ai 2 _ah3 _ _ ai 3 _ (19) By differentiation of Eq. (13a) with respect to time, the acceleration ahi along the ith vertical leg is derived as below: ahi=[£ (e, xg )T ] A + [oiX3 (e, xg, )T ]v , (i = 1, 2, 3), (20a) where, (e xS)T =[-5 (g¡)( ° 190 O) § 185 —I 180 175 —e— Vertical leg h1 —e— Vertical leg h " 3 —e— Horiz ontal leg ^ ontal leg l2 " ontal leg l3 —*— Horiz 0.2 0.8 0 0.2 0.4 0.6 0.8 1 a) Time Is] bj Fig. 3. Length variations in the driving legs; a) length variations in the vertical legs, and b) length variations in the horizontal legs 0.4 0.6 Time [s] E 60 Î"- 40 j? 20 = 0 » -20. 5 ». -40 I -60 > -80 -100 a) —e—Vertical leg h1 —s— Vertical leg h2 —«— Vertical leg h3 150 100 50i 0 o a) -100 —e— Horizontal leg ^ —s— Horizontal leg \2 —*— Horizontal leg l3 0.2 0.8 0.2 0.8 1 Fig. 4. Velocity variations in the driving legs; a) velocity variations in the vertical legs, and b) velocity variations in the horizontal legs 0.4 0.6 Time [s] b) 0.4 0.6 Time [s] The length variations in the vertical legs and in the horizontal legs are shown in Fig. 3. The length ranges of the active legs are all within the design limits. Combined with the structural parameters, the driving legs have no structural interference during movement. The length variations of active legs approximate to the simple harmonic curves and this characteristic benefits the control of the moving platform. The velocity variations of the active legs according to Fig. 3 are shown in Fig. 4, and the corresponding acceleration variations are shown in Fig. 5. Fig. 4a and Fig. 5a indicate that the vertical 800 to £ 600 .c Ol a> 400 a> o 200 tr a) > o 0 c o ro -200 (D