Dynamics of the spherical 3-UPS/S parallel ... - Springer Link

Multibody Syst Dyn (2009) 22: 115–132 DOI 10.1007/s11044-009-9150-x

Dynamics of the spherical 3-UPS/S parallel mechanism with prismatic actuators Stefan Staicu

Received: 24 November 2008 / Accepted: 17 March 2009 / Published online: 8 April 2009 © Springer Science+Business Media B.V. 2009

Abstract Recursive relations in kinematics and dynamics of the symmetric spherical 3-UPS/S parallel mechanism having three prismatic actuators are established in this paper. Controlled by three forces, the parallel manipulator is a 3-DOF mechanical system with three parallel legs connecting to the moving platform. Knowing the position and the rotation motion of the platform, we develop first the inverse kinematics problem and determine the position, velocity, and acceleration of each manipulator’s link. Further, the inverse dynamic problem is solved using an approach based on the principle of virtual work, but it has been verified using the results in the framework of the Lagrange equations with their multipliers. Finally, compact matrix relations and graphs of simulation for the input forces and powers are obtained. Keywords Dynamics modeling · Kinematics · Lagrange equations · Parallel mechanism · Virtual work Nomenclature ak,k−1 , bk,k−1 , ck,k−1 R u1 , u2 , u3 α1i , α2i , α3i (i = A, B, C) α1 , α2 , α3 β ϕk,k−1 ω k,k−1 ω k0 ω˜ k,k−1 εk,k−1 εk0

orthogonal transformation matrices general transformation matrix of the moving platform three orthogonal unit vectors angles giving the position of universal and spherical joints of two platforms Euler angles of successive rotations initial inclination of three legs relative rotation angle of Tk rigid body relative angular velocity of Tk absolute angular velocity of Tk skew-symmetric matrix associated to the angular velocity ω k,k−1 relative angular acceleration of Tk absolute angular acceleration of Tk

S. Staicu () Department of Mechanics, University Politehnica of Bucharest, Bucharest, Romania e-mail: [email protected]

116

ε˜ k,k−1 A rk,k−1 A vk,k−1 A γk,k−1 l0 mp , Jˆp mk Jˆk i i f32 , p32 (i = A, B, C)

S. Staicu

skew-symmetric matrix associated to the angular acceleration εk,k−1 relative position vector of the center of Ak joint relative velocity of the center Ak relative acceleration of the center Ak radius of the circular moving platform mass and tensor of inertia of moving platform mass of Tk rigid body symmetric matrix of tensor of inertia of Tk about the link-frame xk yk zk input forces and powers of three prismatic actuators

1 Introduction Compared with the serial mechanisms, the parallel manipulator is a complex mechanical structure, behaving some special characteristics such as: greater rigidity, potentially higher kinematical precision, stabile capacity, and suitable position of arrangement of actuators. However, they suffer the problems of relatively small useful workspace and design difficulties. Parallel robots can be equipped with hydraulic or prismatic actuators. They have a robust construction and can move bodies of large dimensions with high speeds. Parallel mechanisms can be found in practical applications, in which it is desired to orient a rigid body in space of high speed, such as aircraft simulators [1, 2], positional trackers, and telescopes [3, 4]. More recently, they have been used by many companies in the development of high precision machine tools [5, 6], such as Giddings and Lewis, Hexel, Geodetic, and Toyoda. Recently, considerable efforts have been devoted to the kinematics and dynamic analysis of fully parallel manipulators. Among these, the class of manipulators known as Stewart– Gough-like platform focused great attention (Stewart [2]; Merlet [7]; Parenti Castelli, and Di Gregorio [8]). They are used in flight simulators and more recently for parallel kinematics machines. The prototype of Delta parallel robot (Clavel [9]; Staicu and Carp-Ciocardia [10]; Tsai and Stamper [11]) developed by Clavel at the Federal Polytechnic Institute of Lausanne and by Tsai and Stamper at the University of Maryland as well as the Star parallel manipulator (Hervé and Sparacino [12]) are equipped with three motors and which train on the moving platform in a three-degree-of-freedom general translation motion. Angeles [13], Gosselin and Gagné [14], and Wang and Gosselin [15] analyzed the direct kinematics, dynamics and singularity loci of the Agile Wrist spherical parallel robot with three concurrent actuators. The analysis of parallel manipulators is usually implemented trough analytical methods in classical mechanics, in which projection and resolution of vector equations on the reference axes are written in a considerable number of cumbersome, scalar relations and the solutions are rendered by large scale computations together with time consuming computer codes [16, 17]. In the present paper, a new recursive matrix method is introduced. It has been proved to reduce the number of equations and computation operations significantly by using a set of matrices for kinematics and dynamics models.

Dynamics of the spherical 3-UPS/S parallel mechanism with prismatic

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2 Inverse kinematics A 3-DOF symmetric spherical parallel mechanism (SPM), which can be existent in several applications including machine tools, is considered in the present paper. Since the pneumatic joints can easily achieve high accuracy and heavy loads, many 3-DOF or 6-DOF parallel mechanisms use the actuated prismatic joints. The architecture of the SPM consists of a mobile platform connected first to a fixed base through a central spherical joint and three identical extensible UPS legs, each comprising a prismatic actuator made up of a cylinder and a piston. The upper end of each leg is connected to the moving platform by a spherical joint, whereas the lower end is connected to the fixed base by a universal joint (Fig. 1). Hydraulic or pneumatic actuators can be used to vary the lengths of the prismatic joints and to control the location of the platform. The circle of the moving platform is symbolically represented by three pairs of spherical joints A4 , B4 , C4 , while the circle of the fixed base is represented by another three universal joints located at the centres A1 , B1 , C1 . Both the moving platform DA4 B4 C4 and the fixed base DA1 B1 C1 take the form of a tetrahedron. Overall, there are ten moving links connected together by four spherical, three prismatic and three universal joints. Sometimes we note that this spatial mechanism is not considered a spherical mechanism, because the three legs and the moving platform do not have a common stationary point. The mechanism can be used as an orienting device in which the moving platform is treated as the output link and the three prismatic joints are employed as the input means. For the purpose of analysis, we attach a Cartesian coordinate frame Ox0 y0 z0 (T0 ) to the fixed base with its origin located at the center O, the Oz0 axis perpendicular to the base and the Ox0 axis pointing along the horizontal direction OA1 . Another moving frame Dx3D y3D z3D could be linked in D just at the moving platform. In what follows, we consider that the moving platform is initially located at a central configuration, where the moving platform is not rotated with respect to the fixed base and the mass center G is at an elevation OG = h above the centre of the fixed base (Fig. 2). The mechanism of the robot consists of three chains, including three variable lengths with identical topology, all connecting the fixed base to the moving platform. To simplify the graphical image of the kinematical scheme of the mechanism, in what follows, we will represent the intermediate reference systems by only two axes, so as is used in most of robotics papers [1, 7, 13]. The zk axis is represented, of course, for each component element Tk . It is noted that the relative rotation with angle ϕk,k−1 or the relative translation of the body Tk with the displacement λk,k−1 must always be pointed along the direction of the zk axis. Fig. 1 Symmetric spherical 3-UPS/S parallel mechanism

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S. Staicu

Fig. 2 Kinematical scheme of first leg A of the mechanism

One of these identical active legs A1 A2 A3 A4 (leg A, for example) consists of a fixed Hooke joint, characterized by a mass m1 and a tensor of inertia Jˆ1 , which has the angular A A A A = ϕ˙ 10 and the angular acceleration ε10 = ϕ¨10 , and a moving cylinder of length l2 , velocity ω10 mass m2 and tensor of inertia Jˆ2 , which has a relative rotation about A2 z2A axis with the A A A A A , so that ω21 = ϕ˙ 21 , ε21 = ϕ¨21 . An actuated prismatic joint is as well as a piston angle ϕ21 A A A A = λ˙ A32 , and linked to the A3 x3 y3 z3 frame, having a relative displacement λA32 , velocity v32 A A acceleration γ32 = λ¨ 32 . It has the length l3 , mass m3 , and tensor of inertia Jˆ3 . Finally, a balljoint or a spherical joint is attached to the platform, which can be schematized as a circle of the radius l0 , the mass mp , and the inertia tensor Jˆp . At the central configuration, we also consider that all legs are extended at equal lengths and that the angles of orientation of universal joints and spherical joints are given by 2π , 3 π α2A = α2B = α2C = − , 3 π A B C α3 = α3 = α3 = . 3 α1A = 0,

α1B =

α1C = −

2π , 3 (1)

Assuming that the each leg is connected to the fixed base by universal joint such that it cannot rotate about the longitudinal axis, the orientation of the leg A with respect to the A about the A1 z1A axis, fixed base can be described by two angles, namely a rotation angle ϕ10 A A followed by another rotation of angle ϕ21 about the rotated A2 y2 axis. In the following, we apply the method of successive displacements to geometric analysis of closed-loop chains and we note that a joint variable is the displacement required to move a link from the initial location to the actual position. If every link is connected to least two other links, the chain forms one or more independent closed-loops. The variable angles ϕk,k−1 of rotation about the joint axes zk are the parameters needed to bring the next link from ϕ , for example, a reference configuration to the next configuration. We call the matrix ak,k−1


119

A the orthogonal transformation 3 × 3 matrix of relative rotation with the angle ϕk,k−1 of link A A Tk around zk axis. In the study of the kinematics of robot manipulators, we are interested in deriving a matrix equation relating the location of an arbitrary Tk body to the joint variables. When the change of coordinates is successively considered, the corresponding matrices are multiplied. So, starting from the reference origin O and pursuing the independent legs OA1 A2 A3 , OB1 B2 B3 , OC1 C2 C3 , we obtain the following transformation matrices: ϕ i i θ1 a2α a1α , p10 = p10

ϕ p21 = p21 aβ θ1 θ2 ,

a32 = θ1

(p = a, b, c) (i = A, B, C), (2)

where we denote [18]: ⎡

aσi α

cos ασi ⎣ = − sin ασi 0 ⎡

cos β aβ = ⎣ − sin β 0 ⎡

0 θ1 = ⎣ 0 1

sin ασi cos ασi 0 sin β cos β 0

⎤ −1 0 ⎦, 0

0 1 0

⎡

ϕ pk,k−1

i cos ϕk,k−1 i = ⎣ − sin ϕk,k−1 0

pk0 =

k

pk−s+1,k−s

⎤ 0 0⎦ 1

(σ = 1, 2, 3),

⎤ 0 0⎦, 1 ⎡

0 θ2 = ⎣ −1 0

1 0 0

⎤ 0 0⎦, 1

(3)

⎤ 0 0⎦, 1

i sin ϕk,k−1 i cos ϕk,k−1 0

(k = 1, 2, 3).

s=1

Three independent displacements λA32 , λB32 , λC32 of the active links are the input variables that can control the instantaneous position of the mechanism. But, in the inverse geometric problem, it can be considered that three Euler angles α1 , α2 , α3 of successive relative rotations of the moving platform give the complete position of the mechanism. Since all rotations take place successively about the moving coordinate axes, the resulting rotation matrix is obtained by multiplying three basic rotation matrices: d10 = d1α θ1 , where

⎡

d21 = d2α θ1 θ2 ,

cos αl dlα = ⎣ − sin αl 0

sin αl cos αl 0

⎤ 0 0⎦ 1

d32 = d3α θ2 θ1 θ2

(4)

(l = 1, 2, 3).

(5)

Then the general rotation matrix R of the platform from Ox0 y0 z0 (T0 ) to Dx3D y3D z3D reference system is given by R = d30 = d32 d21 d10 .

(6)

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S. Staicu

We suppose that the angles α1 , α2 , α3 , which are expressed by following wave-form analytical functions π ∗ (l = 1, 2, 3), (7) αl = αl 1 − cos t 3 can describe the absolute motion of the moving platform. C C A A B B The set of nine variables ϕ10 , ϕ21 , λA32 , ϕ10 , ϕ21 , λB32 , ϕ10 , ϕ21 , λC32 will be determined by several vector-loop equations, as follows: A + r10

3

A

T A B ak0 rk+1,k − R T rD 4 = r10 +

k=1

3

B

T B bk0 rk+1,k − R T rD4

k=1 C = r10 +

3

C

T C D ck0 rk+1,k − R T rD4 = r10 ,

(8)

k=1

where i iT = l0 a1α u1 , r10 i = l3 u3 , r43 ⎡ ⎤ 1 u1 = ⎣ 0 ⎦ , 0

i r21 = 0, D r10 = h1 u3 , ⎡ ⎤ 0 u2 = ⎣ 1 ⎦ , 0

i r32 = (l1 + λi32 ) u1 , i

iT iT rD4 = l0 a1α a3α u1 + h2 u3 (i = A, B, C), ⎡ ⎤ ⎡ ⎤ 0 0 −1 0 u3 = ⎣ 0 ⎦ , u˜ 3 = ⎣ 1 0 0 ⎦ . 1 0 0 0

(9)

Actually, these vector equations mean that there is an inverse geometric solution of the first leg A of the manipulator, for example: A D

A + β = l0 cos α2A − uT1 r10 + R T rD 4 cos α1A + α2A l1 + l3 + λA32 cos ϕ21 D

A − uT2 r10 + R T rD 2 sin α1A + α2A ,

A D

A A sin ϕ21 + β = −l0 sin α2A + uT1 r10 + R T rD 4 sin α1A + α2A l1 + l3 + λA32 sin ϕ10 D

A − uT2 r10 + R T rD 2 cos α1A + α2A ,

A D

A A sin ϕ21 + β = uT3 r10 + R T rD 4 . l1 + l3 + λA32 cos ϕ10

(10)

Knowing the rotation motion of the platform by the relations (7), we develop the inA A ,ω k0 , and accelerations verse kinematical problem and determine the absolute velocities vk0 A A γk0 , εk0 of each of the moving links. First, we compute the angular velocities of three legs and the velocities of the joints in terms of the angular velocity of the moving platform 30 = α˙ 1 d32 d21 u3 + α˙ 2 d32 u2 + α˙ 3 u3 . ω p = ω

(11)

The motions of the compounding elements of each leg (leg A, for example) are characterized by following skew symmetric matrices [19]: A A T A = ak,k−1 ω˜ k−1,0 ak,k−1 + ωk,k−1 u˜ 3 ω˜ k0

(k = 1, 2, 3),

(12)


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which are associated to the absolute angular velocities given by the recursive formula A A A ω k0 = ak,k−1 ω k−1,0 + ωk,k−1 u3 .

(13)

A Following relations give the velocities vk0 of joints Ak : A A A A A vk0 = ak,k−1 vk−1,0 + ak,k−1 ω˜ k−1,0 rk,k−1 + vk,k−1 u3 , A vσ,σ −1 = 0

(σ = 1, 2).

(14)

Equations of geometrical constraints (8) will be derivate with respect to time to obtain the following matrix conditions of connectivity established for the characteristic relative velocities of leg A (for example): A

A

A T T T T A A T T T A A T T uj a10 u˜ 3 a21 r43 + ω21 uj a20 u˜ 3 r32 + a32 r43 + v32 uj a20 u1 r32 + a32 ω10 A

= uTj R T ω˜ 30 rD 4

(j = 1, 2, 3),

(15)

where T T T d32 + α˙ 2 d32 u˜ 3 d32 + α˙ 3 u˜ 3 ω˜ p = ω˜ 30 = R R˙ T = α˙ 1 d32 d21 u˜ 3 d21

(16)

denotes the skew-symmetric matrix associated to the absolute angular velocity ω 30 of the A A A , ω21 , v32 as functions moving platform [20]. From (15), we obtain the relative velocities ω10 of angular velocity of the platform. Given from (14), the Jacobian matrix of the robot is a fundamental element for the analysis of singularity loci and workspace of the robot [21–23]. The matrix kinematical relations (15), (16) will be further required in the computation of virtual velocity distribution of the elements of the manipulator. Let us assume now that the manipulator has a first virtual motion determined by the virtual velocities Av = 1, v32a

Bv v32a = 0,

Cv v32a = 0.

(17)

The relations of connectivity (14) express immediately the relative virtual velocities as function of the position of the manipulator. Other two sets of virtual velocities can be obtained Cv Bv = 1, v32c = 1. These virtual velocities are required into the compuif we consider that v32b tation of the virtual power and of the virtual work of all the forces applied to the component elements of the mechanism. A A A , ε21 , γ32 of the elements of first leg A of the mechaAs for the relative accelerations ε10 nism, following other conditions of connectivity are imposed A

A

A T T T T A A T T T A A T T uj a10 u˜ 3 a21 r43 + ε21 uj a20 u˜ 3 r32 + a32 r43 + γ32 uj a20 u1 ε10 r32 + a32

A A A T T T A T A = uTj R T ω˜ 30 ω˜ 30 + ω˙˜ 30 rD 4 − ω10 ω10 uj a10 u˜ 3 u˜ 3 a21 + a32 r43 r32 A

A

A A T T T A A A T T T T A − ω21 ω21 uj a20 u˜ 3 u˜ 3 r32 + a32 r43 − 2ω10 ω21 uj a10 u˜ 3 a21 u˜ 3 r32 + a32 r43 A A T T T A A T T − 2ω10 v32 uj a10 u˜ 3 a21 u1 − 2ω21 v32 uj a20 u˜ 3 u1 ,

(j = 1, 2, 3)

where an useful square matrix is introduced T T T ω˜ 30 ω˜ 30 + ω˙˜ 30 = α¨ 1 d32 d21 u˜ 3 d21 d32 + α¨ 2 d32 u˜ 3 d32 + α¨ 3 u˜ 3 T T T + α˙ 12 d32 d21 u˜ 3 u˜ 3 d21 d32 + α˙ 22 d32 u˜ 3 u˜ 3 d32 + α˙ 32 u˜ 3 u˜ 3

(18)

122

S. Staicu T T T + 2α˙ 1 α˙ 2 d32 d21 u˜ 3 d21 u˜ 3 d32 + 2α˙ 2 α˙ 3 d32 u˜ 3 d32 u˜ 3 T T + 2α˙ 3 α˙ 1 d32 d21 u˜ 3 d21 d32 u˜ 3 .

(19)

A A The accelerations γk0 of the joints Ak and the angular accelerations εk0 are expressed with some relations, founded by the derivatives of (12), (13), (14) A A A A A T = ak,k−1 εk−1,0 + εk,k−1 + ωk,k−1 ak,k−1 ω˜ k−1,0 ak,k−1 u3 , εk0

A A A A A A T ω˜ k0 ω˜ k0 + ε˜ k0 = ak,k−1 ω˜ k−1,0 ω˜ k−1,0 + ε˜ k−1,0 ak,k−1 A A A A T A + ω˜ k,k−1 ω˜ k,k−1 + ε˜ k,k−1 + 2ak,k−1 ω˜ k−1,0 ak,k−1 ω˜ k,k−1 ,

A A A A A A = ak,k−1 γk−1,0 + ak,k−1 ω˜ k−1,0 ω˜ k−1.0 + ε˜ k−1,0 rk,k−1 γk0

(20)

A A T A + 2vk,k−1 ak,k−1 ω˜ k−1,0 ak,k−1 u3 + γk,k−1 u3 , A γσ,σ −1 = 0

(σ = 1, 2).

If other two kinematical chains of the manipulator are pursued, analogous relations can be easily obtained. The matrix relations (18), (19), (20) will be further used for the computation of the wrench of the inertia forces for every rigid component of the robot. The dynamic model of the mechanism would only be established in regard with the complete geometrical analysis and kinematics of the mechanical system. The relations (15) and (18) represent the inverse kinematics model of the symmetric spherical 3-UPS/S parallel mechanism.

3 Equation of motion 3.1 Principle of virtual work The dynamics of parallel manipulators are complicated by the existence of multiple closedloop chains. Difficulties commonly encountered in dynamics modeling of parallel robots include problematic issues such as: complex spatial kinematical structure which possess a large number of passive degrees of freedom, dominance of inertial forces over the frictional and gravitational components, and the problem linked to the solution of the inverse dynamics. In the recent years, many research works have been conducted on the dynamics of parallel structures [24, 25]. In the context of the real-time control, neglecting the friction forces and considering the gravitational effects, the relevant objective of the dynamics is to determine the input forces and powers, which must be exerted by the actuators in order to produce a given trajectory of the end-effector. There are three methods, which can provide the same result concerning these actuating forces. The first one is using the Newton–Euler classic procedure, the second one applies the Lagrange equations and multipliers formalism, and the third one is based on the principle of virtual work [26–29]. A lot of works have focused on the dynamics of Stewart platform. Dasgupta and Mruthyunjaya [16] used the Newton–Euler approach to develop closed-form dynamic equations of Stewart platform, considering all dynamic and gravity effects as well as viscous friction at joints. Tsai [1] presented an algorithm to solve the inverse dynamics for a Stewart platform-type using Newton–Euler equations. This classical approach requires computation of all constraint forces and moments between the links.


123

Geng [30] and Tsai [1] developed Lagrange equations of motion under some simplifying assumptions regarding the geometry and inertia distribution of the manipulator. Liu et al. [31] derived a set of differential equations for the forward dynamics of legs and moving platform, using the Huston form. Based on the theory of screws and on the principle of virtual work, Gallardo et al. [32] allow a calculation of the actuators forces as a function of the external applied forces and the imposed trajectory. Knowing the position and kinematics state of each link as well as the external forces acting on the robot, in the present paper, we apply the principle of virtual work for the inverse dynamic problem in order to establish some definitive recursive matrix relations. The three input forces required in a given rotation motion of the platform will be computed using a matrix recursive procedure. Three independent pneumatic or hydraulic systems A, B, C that generate the input C C A A B B forces f32 = f32 u3 , f32 = f32 u3 , f32 = f32 u3 , which are oriented along the mobile axes C A B A3 z3 , B3 z3 , C3 z3 , control the motion of three moving pistons of the legs. The parallel robot can artificially be transformed in a set of three open chains Ci (i = A, B, C) subject to the constraints. This is possible by cutting each joint for the moving platform, and takes its effect into account by introducing the corresponding constraint conditions. The first and more simplified open tree system could comprise the moving platform only. The wrench of two vectors fk∗ and m ∗k evaluates the influence of the action of the weight mk g and of other external and internal forces applied to the same element Tk of the mechanism fk∗ = 9.81mk ak0 u3 ,

m ∗k = 9.81mk r˜kC ak0 u3 .

(21)

in Now, we compute the force of inertia fkin and the resulting moment of inertia forces m k of an arbitrary rigid body Tk of mass mk with respect to the center of its first joint:

2 fkin = −mk γk0 + ω˜ k0 + ε˜ k0 rkC ,

C k0 + Jˆk ε¯ k0 + ω˜ k0 Jˆk ω k0 . m in k = − mk r˜k γ

(22)

Considering three successive independent virtual motions of the robot, virtual displacements, and velocities should be compatible with the virtual motions imposed by all kinematical constraints and joints at a given instant in time. By intermediate of the complete Jacobian matrix expressed by the conditions of connectivity (15), the absolute virtual vev v ,ω k0 associated with all moving links are related to a set of independent relative locities vk0 v v v v virtual velocities vk,k−1 = vk,k−1 u3 , ω k,k−1 = ωk,k−1 u3 . The fundamental principle of the virtual work [1, 13] states that a mechanism is under dynamic equilibrium if and only if the virtual work developed by all external, internal, and inertia forces vanish during any general virtual displacement, which is compatible with the constraints imposed on the mechanism. Assuming that the frictional forces at the joints are negligible, the virtual work produced by the forces of constraint at the joints is zero. A , for example, inertia forces and Total virtual work contributed by the first active force f32 in in k and by the wrench of known external forces fk∗ , m ∗k can moments of inertia forces fk , m v be written in a compact form, based on the relative virtual angular velocities ωk,k−1 only. Applying the fundamental equations of the parallel robots dynamics established by Staicu [29, 33] the following matrix relation results for the input force of first prismatic actuator:

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S. Staicu

A Av A Av A f32 = uT3 F3A + ω10a M1 + ω21a M2 Bv Bv Cv C Cv C M1 + ω21a M2 + ω10a M1B + ω21a M2B + ω10a

Dv D Dv D Dv D M1 + ω21a M2 + ω32a M3 . + ω10a

(23)

Two recursive relations generate the vectors A T A Fk+1 + ak+1,k , FkA = Fk0 A T A A T A kA = M k+1 k0 M M Fk+1 + ak+1,k + r˜k+1,k ak+1,k ,

(24)

where one denoted A Fk0 = −fkinA − fk∗A ,

A k0 M = −m inA ∗A k −m k .

(25)

The relations (23) and (24) represent the inverse dynamics model of the symmetric spherical 3-UPS/S parallel mechanism. In that follows, we can apply the Newton–Euler procedure to establish the set of analytical equations for each compounding rigid body of a prototype robot in a real application. These equations give all connecting forces in the external and internal joints. Several relations from the general system of equations could eventually constitute verification for the input forces or active torques obtained by the method based on the principle of virtual work. Now, we can calculate the friction forces and the friction torques in the joints, based on the friction coefficients and the maximum of the connecting forces in the joints. We apply again the explicit (23) and (24), where the contribution of the virtual work of friction forces in joints it is added. The new active torques and input forces are compared to the values obtained in the first calculus. Compared with Tsai’s analytical method based on the principle of virtual work [1, 11, 26], the advantages of the present approach are the followings: – Geometrical constraint relations, under matrix form, generate through successive recast the connectivity conditions that will supply all the relative velocities and relative accelerations, which characterize the independent kinematical chains. – The accelerations of the mass centers, the angular accelerations and the twists of the inertia forces are expressed through matrix formulae, which contain the kinematical characteristics of the relative motion of the building elements of the manipulator. – A single matrix relation supplies all the virtual velocities. – The explicit dynamics equation represents a definitive formula, obtained by the transformation of the general expression of the virtual work where the relative virtual velocities only appear, generated by recursive relations. – All intermediate analytical calculations were eliminated and the numerical computation is achieved through the numerical code, for each active force or torque applied by the driving system. 3.2 Equations of Lagrange A solution of the dynamics problem of the 3-UPS/S parallel mechanism can be developed based on the Lagrange equations of second kind for a mechanical system with constraints. The generalized coordinates of the robot are represented by 12 parameters q1 = α1 , q2 = α2 , C C A A B B , q5 = ϕ21 , q6 = λA32 , q7 = ϕ10 , q8 = ϕ21 , q9 = λB32 , q10 = ϕ10 , q11 = ϕ21 , q3 = α3 , q4 = ϕ10 C q12 = λ32 .


125

The Lagrange’s equations with their nine multipliers λ1 , λ2 , . . . , λ9 will be expressed by 12 differential relations 9 d ∂L ∂L = Qk + λs csk − dt ∂ q˙k ∂qk s=1

(k = 1, 2, . . . , 12)

(26)

which contain following 12 generalized forces Q1 = 0, Q2 = 0, Q3 = 0, Q4 = 0, Q5 = 0, C A B , Q7 = 0, Q8 = 0, Q9 = f32 , Q10 = 0, Q11 = 0, Q12 = f32 . Q6 = f32 A number of nine kinematical conditions of constraint are given by the relations (15): 12

csk q˙k = 0

(s = 1, 2, . . . , 9).

(27)

k=1

The components of the general expression of the Lagrange function L = Lp + 3ν=1 (LAν + LBν + LCν ) are expressed as analytical functions of the generalized coordinates and their first derivatives with respect to time: Lp =

1 Tˆ ω Jp ω p − mp g(h1 + h2 cos α1 cos α2 ), 2 p

Li1 =

1 iT î i ω J ω , 2 10 1 10

Li2 =

1 iT î i T Ci ω J ω − mi2 g uT3 p20 r2 , 2 20 2 20

Li3 =

T i

1 i iT i 1 iT î i iT i Ci T Ci m3 v30 v30 + ω 30 J3 ω 30 + mi3 v30 ω˜ 30 r3 − mi3 g uT3 p20 r32 + p30 r3 , 2 2

i = (A, B, C),

(28)

p = (a, b, c).

The joint’s velocities, the angular velocities, the skew-symmetric matrices associated to the angular velocities and the first derivatives of orthogonal matrices pk,k−1 are expressed as follows: i = 0, v10

i v20 = 0,

i i = ϕ˙ 10 u3 , ω 10

i i i v30 = p32 ω˜ 20 r32 + λ˙ i32 u3 ,

i i i ω 20 = ϕ˙ 10 p21 u3 + ϕ˙ 21 u3 ,

i i i = ϕ˙ 10 p32 p21 u3 + ϕ˙ 21 p32 u3 , ω 30 i i = ϕ˙ 10 u˜ 3 , ω˜ 10

i i i T ω˜ 20 = ϕ˙ 10 p21 u˜ 3 q21 + ϕ˙ 21 u˜ 3 ,

i i i T T T = ϕ˙ 10 p32 p21 u˜ 3 p21 p32 + ϕ˙ 21 p32 u˜ 3 p32 , ω˜ 30 i u˜ T3 pk,k−1 , p˙ k,k−1 = ϕ˙ k,k−1

∂pk,k−1 = u˜ T3 pk,k−1 , i ∂ϕk,k−1

i T T p˙ k,k−1 = ϕ˙ k,k−1 pk,k−1 u˜ 3 ,

T ∂pk,k−1 i ∂ϕk,k−1

T = pk,k−1 u˜ 3 .

(29)

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Now, we can calculate the partial derivatives of the Lagrange functions ∂Lp = mp gh2 sin α1 cos α2 , ∂α1

∂Lp T T ˆ = uT3 R21 R32 Jp ω p, ∂ α˙ 1

∂Lp T T ˆ Jp ω = mp gh2 cos α1 sin α2 + α˙ 1 uT3 R21 u˜ 3 R32 p, ∂α2 ∂Lp T ˆ = uT3 R32 p, Jp ω ∂ α˙ 2 ∂Lp T T T = α˙ 1 uT3 R21 R32 u˜ 3 Jˆp ω p + α˙ 2 uT3 R32 u˜ 3 Jˆp ω p, ∂α3 ∂Lp = uT3 Jˆp ω p, ∂ α˙ 3 ∂Li1 = 0, i ∂ϕ10

∂Li1 i = uT3 Jˆ1i ω 10 , i ∂ ϕ˙ 10

∂Li2 T T Ci = −mi2 g uT3 p10 u˜ 3 p21 r2 , i ∂ϕ10

∂Li2 T î i J2 ω = uT3 p21 20 , i ∂ ϕ˙ 10

∂Li2 i i T T = ϕ˙ 10 uT3 p21 u˜ 3 Jˆ2i ω 20 − mi2 g uT3 p20 u˜ 3 r2Ci , i ∂ϕ21 ∂Li2 i = uT3 Jˆ2i ω 20 , i ∂ ϕ˙ 21

(30)

i

∂Li3 T T T Ci = −mi3 g uT3 p10 u˜ 3 p21 r3 , r32 + p32 i ∂ϕ10 i

∂Li3 i Ci i i T i T T T = mi3 uT3 p21 r˜32 p32 r3 + uT3 p21 p32 mi3 r˜3Ci v30 + Jˆ3i ω 30 v30 + ω˜ 30 , i ∂ ϕ˙ 10 i

i Ci i

∂Li3 i i i Ci i i T T T T = mi3 ϕ˙ 10 uT3 p21 u˜ 3 r˜32 p32 r3 + ϕ˙ 10 uT3 p21 u˜ 3 p32 30 v30 + ω˜ 30 m3 r˜3 v30 + Jˆ3i ω i ∂ϕ21 i

T T Ci − mi3 g uT3 p20 u˜ 3 r32 + p32 r3 , i

i Ci i

∂Li3 i i Ci i T T = mi3 uT3 r˜32 p32 r3 uT3 p32 30 v30 + ω˜ 30 m3 r˜3 v30 + Jˆ3i ω , i ∂ ϕ˙ 21 i

∂Li3 iT i Ci T T = mi3 ω 20 u˜ 1 p32 r3 − mi3 g uT3 p20 u1 , v30 + ω˜ 30 ∂λi32 i

∂Li3 i Ci = mi3 uT3 v30 + ω˜ 30 r3 . ∂ λ˙ i32 } (k = A long and tedious calculus of the derivatives with respect to time dtd { ∂∂L q˙k 1, 2, . . . , 12) of some above functions leads to an algebraic system of 12 relations. In the direct or inverse dynamics problem, after elimination of the nine multipliers, finally we obtain


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C A B same expressions (23) for the six input forces f32 , f32 , f32 required by the three prismatic actuators. As applications let us consider a manipulator, which has the following characteristics

π π π , α2∗ = , α3∗ = , 12 18 36 m2 = 2.5 kg, m3 = 1.5 kg, mp = 5 kg, m1 = 0.35 kg, √ l1 = 0.25 m, l = 0.75 m, l0 = OA1 = l/ 3, α1∗ =

l2 = 1.2 m, cos β = l0 /(l1 + l3 ),

l3 = 1.25 m,

l4 = GA4 = l0 ,

h = OG = (l1 + l3 ) sin β,

OD = h1 = DG = h2 = h/2,

t = 3 s.

4 Simulation procedure The procedure for solving the inverse dynamics of the 3-UPS/S parallel robot by using the principle of virtual work can be summarized in several basic steps. 1. For a period of t = 3 second, it is assumed that the time-history of the moving platform is specified in terms of its orientation about the fixed joint D from (7). The relations (8) i i i , ϕ21 , ϕ32 , (i = A, B, C). give the evolution of joint variables ϕ10 2. Using the relations (2), (3), (4), and (6) we compute the transformation matrices of three i i i i i i i i i , p21 , p32 , p20 = p21 p10 , p30 = p32 p20 (p = a, b, c) and the general legs A, B, C: p10 rotation matrix R. 3. Determine the velocities and accelerations of all links by performing the inverse kinematics analysis in terms of prescribed angular velocities α˙ 1 , α˙ 2 , α˙ 3 and angular accelerations α¨ 1 , α¨ 2 , α¨ 3 of the moving platform. Specifically, for each leg, from the conditions of i i i , ω21 , v32 and the connectivity (15), (18), we compute the relative angular velocities ω10 i i i relative angular accelerations ε10 , ε21 , γ32 . Cv Av Bv = 1, v32a = 0, v32a = 0, for example, we 4. Using same (15), where we introduce v32a compute the virtual characteristic velocity of each element of the robot. Other sets of Cv Bv = 1, v32c = 1. virtual velocities are obtained if we consider successively that v32b 5. We compute the velocity and the acceleration of center of mass as well as the angular velocity and the angular acceleration of each link using (13), (14), and (20). 6. Decompose artificially the robot in several open-loop chains by cutting open at the moving revolute joints A3 , B3 , C3 . A more simplified system is just the moving platform. 7. For each link and platform, we determine the inertia force (22) and the resulting force (excluding the actuator force) exerted to the rigid body Tk , from recursive (24). 8. For each link and platform, we determine the moment of inertia forces (22) and the resulting moment (excluding the actuator torque) exerted at the joint Ak , from recursive (24). A A and the power p32 from (23). Analogous relations give the 9. We find the input force f32 C C B B input forces f32 , f32 and powers p32 , p32 of the other two actuators. Based on the algorithm derived from above equations, a computer program was developed to solve the inverse dynamics of the 3-UPS/S symmetric spherical parallel mechanism, using the MATLAB software. To validate the dynamics modeling, it is assumed that

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Fig. 3 Input forces of three actuators

Fig. 4 Powers of three actuators

the platform starts at rest from a central configuration and rotates about one of three orthogonal directions. Furthermore, at the initial location, the moving platform is assumed to be located as follows: t = 0, αl = 0 (l = 1, 2, 3). Assuming that there are not external forces and moments acting on the moving plati i i = v32 f32 (i = A, B, C) form, the time-history evolution of the input forces and powers p32 required by three pneumatic or hydraulic active systems are shown for a period of three second of platform’s rotation. The following examples are solved to illustrate the algorithm. In a first example, the moving platform rotates about z3D axis with a variable angular acceleration α¨ 3 while all the other positional parameters are held equal to zero. As can be seen from Fig. 3 and Fig. 4, it is proved to be true that all input forces and powers are permanently equal to one another. If we consider the rotation motion of the platform about the horizontal x3D direction, the input forces and the powers are calculated by the program and plotted versus time. We remark a permanent equality for the input forces (Fig. 5) or the powers (Fig. 6) of first and second actuators.


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For the third motion, the platform accomplishes a general rotation about the fixed joint D. From Fig. 7 and Fig. 8, it is remarked that, during the rotation motion of the platform, the actuators B and C only are working with positive values of the absorbed powers.

5 Conclusions Some dynamical models based on the Lagrange formalism neglect the weight of intermediate bodies and take into consideration the active forces or moments only and the wrench of applied forces on the moving platform. The number of relations given by the Lagrange formalism is equal to the total number of the position variables and Lagrange multipliers inclusive. However, the analytical calculations involved in these equations present an elevated risk of errors.

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The commonly known Newton–Euler method, which takes into account the free-bodydiagrams of the mechanism, leads to a large number of equations with unknowns including also the actuating and connecting forces in the joints. Within the inverse kinematics analysis, the conditions of connectivity (15), (18) that give in real-time the relative velocity and acceleration of each element of the parallel robot have been established in the present paper. The dynamics model takes into consideration the mass, the tensor of inertia and the action of weight and inertia force introduced by each element of the manipulator. Based on the principle of virtual work, the new approach is far more efficient, can eliminate all forces of internal joints, and establishes a direct determination of the time-history evolution of active forces and powers required by the three actuators. The recursive matrix relations (23), (24) represent a set of explicit equations of the dynamic simulation, and in a context of automatic command, can easily be transformed into a


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robust model for computerized control of a 3-UPS/S symmetric spherical parallel mechanism.

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