Matrix modeling of inverse dynamics of spatial and

Multibody Syst Dyn (2012) 27:239–265 DOI 10.1007/s11044-011-9281-8

Matrix modeling of inverse dynamics of spatial and planar parallel robots Stefan Staicu

Received: 5 November 2009 / Accepted: 17 October 2011 / Published online: 11 November 2011 © Springer Science+Business Media B.V. 2011

Abstract Recursive matrix relations for kinematics and dynamics analysis of two known parallel mechanisms: the spatial 3-PRS and the planar 3-RRR are established in this paper. Knowing the motion of the platform, we develop first the inverse kinematical problem and determine the positions, velocities, and accelerations of the robot’s elements. Further, the inverse dynamic problem is solved using an approach based on the principle of virtual work, and the results can be verified in the framework of the Lagrange equations with their multipliers. Finally, compact matrix equations and graphs of simulation for power requirement comparison of each of three actuators in two different actuation schemes are obtained. For the same evolution of the moving platform, the power distribution upon the three actuators depends on the actuating configuration, but the total power absorbed by the set of three actuators is the same, at any instant, for both driving systems. The study of the dynamics of the parallel mechanisms is done mainly to solve successfully the control of the motion of such robotic systems. Keywords Dynamics modeling · Kinematics · Lagrange equations · Parallel mechanism · Virtual work Nomenclature ak,k−1 , bk,k−1 , ck,k−1 R θ1 , θ2 , θ3 u1 , u2 , u3 l1 , l2 θ ϕk,k−1 ω k,k−1 ω k0

orthogonal rotation matrices general rotation matrix of moving platform three constant orthogonal matrices three right-handed orthogonal unit vectors length of two links of each leg angle of inclination of three sliders relative rotation angle of Tk rigid body relative angular velocity of Tk absolute angular velocity of Tk

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

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ω˜ k,k−1 εk,k−1 εk0 ε˜ k,k−1 A rk,k−1 A vk,k−1 A γk,k−1

rkC mk , Jˆk mp , Jˆp C A B , f10 , f10 , mA10 , mB10 , mC10 f10 mA21 , mB21 , mC21 , mA32 , mB32 , mC32

S. Staicu

skew-symmetric matrix associated to the angular velocity ω k,k−1 relative angular acceleration of Tk absolute angular acceleration of Tk skew-symmetric matrix associated to the angular acceleration εk,k−1 relative position vector of the center Ak of joint relative velocity of the center Ak relative acceleration of the center Ak position vector of the mass center of Tk rigid body mass and symmetric matrix of tensor of inertia of Tk about the link-frame xk yk zk mass and central tensor of inertia of moving platform forces and torques of fixed actuators torques of mobile actuators

1 Introduction Parallel robots are closed-loop structures presenting very good potential in terms of accuracy, rigidity and ability to manipulate large loads. Recently, parallel mechanisms have gained greater attention within the research community as their advantages become better known. In general, these mechanisms consist of two main bodies coupled via numerous legs acting in parallel. One body is designated as fixed and is called base, while the other is regarded as movable and hence is called moving platform of the manipulator. Several legs or limbs, made up as serial chains, connect the moving platform to the fixed frame. The links of the robot are connected one to the other by spherical joints, universal joints, revolute joints or prismatic joints. The number of actuators is typically equal to the number of degrees of freedom such that each leg is controlled at or near the fixed base. Compared with serial robots, the following are the potential advantages of parallel architectures: higher kinematical accuracy, lighter weight and better structural rigidity, stable capacity and suitable position of arrangement of actuators, low manufacturing cost and simple control algorithms. But from an application point of view, a limited workspace and complicated singularities are two major drawbacks of parallel manipulators. Parallel manipulators can be equipped with revolute or prismatic actuators. They have a robust construction and can move bodies of large dimensions with high velocities and accelerations. That is the reason why the devices, which produce translation or spherical motion to a platform, technologically are based on the concept of parallel manipulators [1]. Most three DOF mechanisms exist in one of three of the following classes: spherical, translational or spatial. A translation device is capable of moving in Cartesian coordinates and is suitable for pick-and-place applications. Spherical mechanisms are strictly for pointing applications such as orienting a camera. The final class is the class of spatial mechanisms, which include a combination of both rotational and translational degrees of freedom. Over the past two decades, parallel manipulators have received more attention from researches and industries. Important companies such as Giddings & Lewis, Ingersoll, Hexel and others have developed them as high precision machine tools. Accuracy and precision in the direction of the tasks are essential since the positioning errors of the tool could end in costly damage.

Matrix modeling of inverse dynamics of spatial and planar parallel

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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 platform focused great attention (Stewart [2]; Merlet [3]; Parenti Castelli and Di Gregorio [4]). They are used in flight simulators and more recently for Parallel Kinematics Machines. The prototype of Delta parallel robot (Clavel [5]; Staicu [6]; Tsai and Stamper [7]) 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 [8]) are equipped with three motors, which train on the mobile platform in a three-degreeof-freedom general translation motion. Angeles [9], Wang and Gosselin [10], Staicu [11] analyzed the kinematics, dynamics and singularity loci of Agile Wrist spherical robot with three actuators. The analysis of parallel manipulators is usually implemented through 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. In the present paper, a recursive matrix method, already implemented in the inverse dynamics of parallel robots, is applied to the analysis of planar and spatial three-degrees-offreedom mechanisms. 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.

2 Spatial 3-PRS parallel manipulators The 3-PRS mechanism has been studied by a variety of researchers [12–14]. This parallel robot consists of a moving platform, a fixed base and three limbs of identical structure. The potential application as a positioning device of the tool in a new parallel kinematics machine for high precision blasting attracted a scientific and practical interest to this manipulator type. The robot motion in the translation and orientation degrees of freedom is interconnected. This makes the investigations more complicated and this problem does not exist in the manipulators with a different structure. The inclined version of the 3-PRS mechanism, first introduced by Carretero et al. [15], consists of three identical legs formed by prismatic joints followed by revolute joints and three passive spherical joints. Another similar architecture was introduced by Tsai et al. [16], where the lines of action of prismatic joints are all parallel to each other and perpendicular to the base platform. A micro-level version of the 3-PRS robot was presented by Merlet [17], where the fixed-length link leans outwards, toward the extremity of the end-effectors. Li and Xu [18] analyzed kinematics equations and the dynamics for the case when the manipulator motion is determined by two degrees of orientation freedom and one degree of vertical translation. Having a closed-loop structure, the spatial 3-PRS parallel manipulator is a special symmetrical mechanism composed of three kinematical chains of variable length with identical topology, all connecting the fixed base to the moving platform. The extensible leg connects the moving platform by spherical joint and the base by means of prismatic joint, the axis of which is inclined from the base platform. Ball screw or pneumatic jacks can be used to vary the lengths of the prismatic joints and to control the location of the platform. Fixed actuated prismatic joints or mobile revolute joints can drive the manipulator.

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Fig. 1 The 3-PRS parallel manipulator

2.1 Kinematics analysis The manipulator consists of the fixed base and the upper platform A3 B3 C3 that are √ two , r and the lengths of the sides L = l equilateral triangles characterized by the radii l 0 0 3, √ l = r 3, respectively. Grübler mobility equation predicts that the device has certainly three degrees of freedom. Practically, the degree-of-freedom value F = 3 of the spatial mechanism is equal to the degrees of freedom associated with all the moving links ν = 12 minus the total number of independent constraint relations l = 9 imposed by the joints. In a first kind of the robot (PRS) we consider the moving platform as the output link and the links A1 A2 , B1 B2 , C1 C2 as the input links. Thus, all actuators can be installed on the fixed base. In the second configuration (PRS) each intermediate revolute joint is an actively controlled joint. Having a common point of intersection, the lines of action of each of three prismatic joints may be inclined from the vertical axis by a fixed angle θ as architectural parameter. These lines and the revolute joint axes are mutually perpendicular. Finally, the revolute joint of each leg is separated from the spherical joint attached to the platform by a fixed-length link. In this configuration, any translation along the vertical axis is limited by the constraints and therefore only finite displacements along this axis are obtained. For the purpose of analysis, a Cartesian frame x0 y0 z0 (T0 ) is attached to the fixed base with its origin located at triangle center O, the z0 axis perpendicular to the base. Another mobile reference frame xG yG zG is attached to the moving platform. The origin of this coordinate central system is located just at the center G of the moving triangle. The kinematical constraints of each leg limit the motion of spherical joint of each limb to a plane perpendicular to the axis of the revolute joint and containing the spherical joint. In the symmetrical version of the manipulator, the first leg A is typically contained within the Ox0 z0 vertical plane, whereas the remaining two legs B, C make angles αB = 120°, αC = −120° respectively, with the first leg (Fig. 1). To simplify the graphical image of the kinematical scheme of the mechanism, in the follows we will represent the intermediate reference systems by only two axes, so as used in most of books [1, 3, 9]. The zk axis is represented for each element Tk . It is noted that the relative rotation with ϕk,k−1 angle or relative translation of Tk body with λk,k−1 displacement must always be pointing about or along the direction of zk axis. In the following we consider that the moving platform is initially located at a central configuration, where the platform is not rotated with respect to the fixed base and the mass center G is located at an elevation OG = h above the origin O of fixed frame. A complete


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description of the position and orientation of the moving platform with respect to the reference frame requires generally six variables: the coordinates x0G , y0G , z0G of the mass center G and a set of three Euler’s angles φ1 , φ2 , φ3 associated with three successive rotations. Since all rotations take place successively about the moving coordinate axes, the general rotation matrix R = d32 d21 d10 of the moving platform from Ox0 y0 z0 to GxG yG zG reference system is obtained by multiplying three transformation matrices d10 = a1 θ1 , where

⎡

d21 = a2 θ1 θ2 ,

⎤ 0 0 −1 θ1 = ⎣ 0 1 0 ⎦ , 1 0 0 ⎡ cos φk sin φk ak = ⎣ − sin φk cos φk 0 0

d32 = a3 θ2 θ1 θ2

⎤ 0 1 0 θ2 = ⎣ −1 0 0 ⎦ , 0 0 1 ⎤ 0 0 ⎦ (k = 1, 2, 3). 1

(1)

⎡

(2)

However, the mechanism is only a 3-DOF device, therefore only three of them are independent. The defined independent degrees of freedom for the mechanism are a translation along the z0 axis and the rotations around two orthogonal moving axes. In order to obtain this desired motion, the manipulator may be forced to make two orthogonal parasitic translations and a parasitic rotation. Here, z0G , φ1 and φ2 are chosen as the independent variables and x0G , y0G , φ3 are parameters of parasitic motions. The three parasitic motions from the six commonly known motions of the moving platform are permanently dependent on three independent variables. One of three active legs (for example leg A) consists of a prismatic joint as well as a piston linked at the x1A y1A z1A moving frame, having a translation with the displacement λA10 , A A = λ˙ A10 and the acceleration γ10 = λ¨ A10 . It has the length l1 and the mass m1 . the velocity v10 An intermediate link of length l2 , mass m2 and tensor of inertia Jˆ2 has a relative rotation A A A , the angular velocity ω21 = ϕ˙ 21 and the angular acceleration about z2A axis with the angle ϕ21 A A = ϕ¨21 . Finally, a spherical joint is introduced at a planar moving platform, which is ε21 schematized as an equilateral triangle with edge l, mass mp and tensor of inertia Jˆp (Fig. 2). At the central configuration, we also consider that all legs are initially extended at equal length λ0 + l1 = [h − l2 sin(θ − β)]/ cos θ and that the angles of orientation of the legs are given by 1 2π 2π 2 , cos θ = √ , , αC = − , sin θ = αB = αA = 0, 3 3 3 3 (3) −1 h − (λ0 + l1 ) cos θ β = θ − tg . (λ0 + l1 ) sin θ − r 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 at 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 ϕ , link from a reference configuration to the next configuration. We call the matrix ak,k−1 for example, the orthogonal transformation 3 × 3 matrix of relative rotation with the anA of link TkA around zkA axis. In the study of the kinematics of robot manipulators, gle ϕk,k−1

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

we are interested in deriving a matrix equation relating the location of an arbitrary 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: p10 = aθ aαi , where we denote [19]: ⎡ cos αi sin αi aαi = ⎣ − sin αi cos αi 0 0 ⎡

cos θ aθ = ⎣ 0 sin θ ⎡

ϕ pk,k−1

0 1 0

ϕ p21 = p21 aβ θ1 θ2 θ3

⎤ 0 0⎦, 1

⎤ − sin θ 0 ⎦, cos θ

i cos ϕk,k−1 i ⎣ = − sin ϕk,k−1 0

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

⎡

cos β aβ = ⎣ − sin β 0 ⎡

−1 0 θ3 = ⎣ 0 −1 0 0

i sin ϕk,k−1 i cos ϕk,k−1 0

⎤ 0 0⎦, 1

sin β cos β 0

⎤ 0 0⎦, 1

⎤ 0 0⎦, 1

pk0 =

k

(4)

pk−s+1,k−s

(5)

(k = 1, 2).

s=1

Three independent displacements λA10 , λB10 , λC10 of the active links are the joint variables that 10 = [λA10 λB10 λC10 ]T of the instantaneous pose of the mechanism in give the input vector λ the first study configuration. But in the inverse geometric problem it can be considered that the position of the mechanism is completely given through the coordinate z0G of the mass center G and two Euler angles φ1 , φ2 of rotation about two mobile axes xG , yG , respectively. Further, we suppose that the following three analytical functions can describe the general absolute motion of the moving platform: π π π φ1 = φ1∗ 1 − cos t , φ2 = φ2∗ 1 − cos t . (6) z0G = h + z0G∗ 1 − cos t , 3 3 3


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C A B Six independent variables λA10 , ϕ21 , λB10 , ϕ21 , λC10 , ϕ21 and the parameters φ3 , x0G , y0G will be determined by several vector-loop equations as follows: A + r10

2

A

T A B ak0 rk+1,k − R T rG3 = r10 +

2

k=1

B

T B bk0 rk+1,k − R T rG3

k=1 C = r10 +

2

C

T C ck0 rk+1,k − R T rG 3 = r0G

(7)

k=1

where

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0 0 −1 0 u1 = ⎣ 0 ⎦ , u2 = ⎣ 1 ⎦ , u3 = ⎣ 0 ⎦ , u˜ 3 = ⎣ 1 0 0 ⎦ , 0 0 1 0 0 0 i i i i T r10 = λ0 + λi10 p10 u3 , r21 = l1 u3 , r32 = −l2 u2 , rG3 = raαiT u1

(8)

(i = A, B, C). Actually, these vector equations mean that there is only one inverse geometric solution for the spatial manipulator: i + β − θ = x0G + q1i / cos αi = y0G + q2i / sin αi λ0 + l1 + λi10 sin θ − l2 cos ϕ21 (9) i + β − θ = z0G + q3i , λ0 + l1 + λi10 cos θ − l2 sin ϕ21 with the notations i

i

q1i = xG3 cos φ2 cos φ3 − yG3 cos φ2 sin φ3 , i

q2i = xG3 (sin φ1 sin φ2 cos φ3 + cos φ1 sin φ3 )

(10)

i

+ yG3 (− sin φ1 sin φ2 sin φ3 + cos φ1 cos φ3 ), i

i

q3i = xG3 (− cos φ1 sin φ2 cos φ3 + sin φ1 sin φ3 ) + yG3 (cos φ1 sin φ2 sin φ3 + sin φ1 cos φ3 ). We can obtain the following significant functions from (9): tg φ3 = −

sin φ1 sin φ2 , cos φ1 + cos φ2

x0G = 0.5r(sin φ1 sin φ2 sin φ3 − cos φ1 cos φ3 + cos φ2 cos φ3 ),

(11)

y0G = r cos φ2 sin φ3 . So, for the 3-PRS mechanism, the z0G coordinate is the only completely independent variable with other five pose parameters. Now, we develop the inverse kinematics problem and determine the velocities and accelerations of the robot, supposing that the motion of the moving platform is known. First, we C C A B A B , v10 , v10 and the angular velocities φ˙ 3 , ω21 , ω21 , ω21 compute the linear velocities x˙0G , y˙0G , v10 in terms of the angular velocities φ˙ 1 , φ˙ 2 and the vertical velocity z˙ 0G of the moving platform. The motion of the compounding elements of the leg A, for example, are characterized by the following skew-symmetric matrices: A A T A = ak,k−1 ω˜ k−1,0 ak,k−1 + ωk,k−1 u˜ 3 , ω˜ k0

A A ωk,k−1 = ϕ˙ k,k−1 ,

(12)

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

(k = 1, 2).

(13)

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A The following relations give the velocities vk0 of the 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 ω10 = 0,

A v21 = 0.

(14)

The vector equations of geometrical constraints (7) can be differentiated with respect to time to obtain the following significant matrix conditions of connectivity: A T T A T v10 uj a10 u3 + ω21 l3 uTj a20 u1 = uTj r˙ 0 + uTj R T ω˜ p rG3 G

A

(j = 1, 2, 3),

(15)

where T T T ω˜ p = φ˙ 1 d32 d21 u˜ 3 d21 d32 + φ˙ 2 d32 u˜ 3 d32 + φ˙ 3 u˜ 3

(16)

denotes the skew-symmetric matrix associated to the known angular velocity of moving platform ω p = φ˙ 1 d32 d21 u3 + φ˙ 2 d32 u3 + φ˙ 3 u3 .

(17)

If the other two kinematical chains of the robot are pursued, analogous relations can be easily obtained. From these equations, we obtain the complete Jacobian matrix of the manipulator. This matrix is a fundamental element for the analysis of the robot workspace and the particular configurations of singularities where the spatial manipulator becomes uncontrollable [20]. C A B The rotations angles ϕ21 , ϕ21 , ϕ21 of the intermediate links are the joint variables that can C T A B give the input vector ϕ21 = [ϕ21 ϕ21 ϕ21 ] of the instantaneous position of the mechanism in the second actuation scheme. The matrix kinematical relations (12), (13), (14), and (15) will be further required in the computation of virtual velocity distribution of the elements of the manipulator. Let us assume that the manipulator has successively two virtual motions determined by the linear Cv Cv Av Bv Av Bv velocities v10a = 1, v10a = 0, v10a = 0 or the angular velocities ω21a = 1, ω21a = 0, ω21a = 0. The characteristic virtual velocities are expressed as functions of the position of the mechanism by the general kinematical constraints equations (15). These virtual velocities are required into the computation of the virtual power and of the virtual work of all the forces applied to the component elements of the mechanism. C A B As for the linear accelerations x¨0G , y¨0G , γ10 , γ10 , γ10 and the angular accelerations C A B ¨ φ3 , ε21 , ε21 , ε21 of the robot, the derivatives with respect to time of (15) give these other conditions of connectivity [21]:

A G A T T A T A A T γ10 uj a10 u3 + ε21 l3 uTj a20 u1 = uTj r¨ 0 + uTj R T ω˜ p ω˜ p + ω˙˜ p rG3 − ω21 ω21 l3 uTj a20 u2 (j = 1, 2, 3)

(18)

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

(19)


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A A The following recursive relations give the angular accelerations εk0 and the accelerations γk0 of joints Ak : A A A A A T εk0 = ak,k−1 εk−1,0 + εk,k−1 u3 + ωk,k−1 ak,k−1 ω˜ k−1,0 ak,k−1 u3 , A A A A A A T A A ω˜ k0 + ε˜ k0 = ak,k−1 ω˜ k−1,0 ω˜ k−1,0 + ε˜ k−1,0 + ωk,k−1 ωk,k−1 u˜ 3 u˜ 3 ak,k−1 ω˜ k0 A A A T + εk,k−1 u˜ 3 + 2ωk,k−1 ak,k−1 ω˜ k−1,0 ak,k−1 u˜ 3 , A A A A A A γk0 = ak,k−1 γk−1,0 + ak,k−1 ω˜ k−1,0 ω˜ k−1,0 + ε˜ k−1,0 rk,k−1 A A T A + 2vk,k−1 ak,k−1 ω˜ k−1,0 ak,k−1 u3 + γk,k−1 u3 A ε10

= 0,

A γ21

(20)

(k = 1, 2),

= 0.

The matrix relations (18), (19), and (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. 2.2 Dynamics equations The kinematics and the dynamics of parallel structures have been studied extensively during the last two decades. When a good dynamic performance and a higher accuracy in positioning of the moving platform under large load are required, the dynamic model of the robot is important for the automatic control. The inverse dynamic model is of great significance in the optimization of sectional parameters of components and estimation of servomotors parameters. The dynamics of parallel manipulators is complicated by the existence of multiple closedloop chains. Difficulties commonly encountered in dynamics modeling of parallel robots include problematic issues such as: complicated spatial kinematical structure which possesses 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 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 torques or forces, which must be exerted by the actuators in order to produce a given trajectory of the end-effector. Up to now, several methods have been applied to formulate the dynamics of parallel mechanisms, which could provide the same results concerning these actuating torques or forces. The first one is using the Newton–Euler classical procedure [22–24], the second one applies the Lagrange’s equations and multipliers formalism [25, 26] and the third one is based on the principle of virtual work [1, 9, 27–29]. Kane and Levinson [30] had derived important recursive relations concerning the equilibrium of generalized forces that are applied to a spatial manipulator, commonly known as Kane’s equations. Much work has focused on the dynamics of Stewart platform. Dasgupta and Mruthyunjaya [23] 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 platformtype using Newton–Euler equations. This commonly known approach requires computation of all constraint forces and moments between the links. Geng [25] and Tsai [1] developed Lagrange equations of motion under some simplifying assumptions regarding the geometry and inertia distribution of the manipulator.

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2.2.1 Principle of virtual work 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 and powers required in a given motion of the platform will be computed using a recursive procedure. A A B B Three independent mechanical systems, acting with the forces f10 = f10 u3 , f10 = f10 u3 , C C A B C f10 = f10 u3 along the directions z1 , z1 , z1 , or three electric motors A2 , B2 , C2 that generate the torques m A21 = mA21 u3 , m B21 = mB21 u3 , m C21 = mC21 u3 oriented about mobile horizontal axes, control the motion of the moving platform. 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 force of inertia and the resulting moment of the forces of inertia of an arbitrary rigid body TkA , for example, A A A CA inA A fk0 = −mAk γk0 + ω˜ k0 ω˜ k0 + ε˜ k0 rk , (21) A CA A inA A A A A k0 m k0 = − mk r˜k γk0 + Jˆk εk0 + ω˜ k0 JˆkA ω are determined with respect to the center of joint Ak . On the other hand, the wrench of two A ∗A and of other vectors fk∗A and m k evaluates the influence of the action of the weight mk g A external and internal forces applied to the same element Tk of the manipulator: fk∗A = mAk gak0 u3 ,

A CA m ∗A 3 k = mk g r˜k ak0 u

(k = 1, 2).

(22)

The fundamental principle of the virtual work 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 all forces of constraint at the joints is zero. Applying the definitive form of fundamental equations of parallel robots dynamics [31, 32], the following compact matrix relations results:

A Av A Bv B Cv C f10 M2 + ω21a M2 + ω21a M2 = uT3 F1A + ω21a Dv D Dv D Dv D Dv D Dv D Dv D M2 + ω32a M3 + v10a F1 + v21a F2 + v32a F3 + ω10a M1 + ω21a (23) for the input force of first active prismatic joint and

A Av A Bv B Cv C 2 + v10a F1 + v10a F1 + v10a F1 mA21 = uT3 M Dv D Dv D Dv D Dv D Dv D Dv D M1 + ω21a M2 + ω32a M3 + v10a F1 + v21a F2 + v32a F3 + ω10a

(24)

for the input torque of first active revolute joint. Two recursive relations generate the vectors A T A A T A A T A k+1 kA = M k0 Fk+1 M Fk+1 + ak+1,k , M + ak+1,k + r˜k+1,k ak+1,k FkA = Fk0

(25)

where one denoted A inA Fk0 = −fk0 − fk∗A ,

A k0 M = −m inA ∗A k0 − m k .

(26)

The relations (23), (24), (25), and (26) represent the inverse dynamics model of the 3-PRS spatial parallel robot. These explicit equations are definitively written in a recursive comv v and ωk.k−1 . The pact form, only based on the relative virtual velocities of intensities vk,k−1 various dynamical effects, including the Coriolis and centrifugal forces coupling and the


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gravitational actions are considered. This dynamics model is successfully expected to be deployed for robotic control. The above compact relations can be also applied to calculate any internal joint force or joint torque by considering several fictitious displacements or rotations in each joint of the manipulator. The present method is general and can easily be extended to the mechanical analysis of other types of parallel manipulator. In what 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 equations (23), (24), (25), and (26), where the contribution of the virtual work of friction forces in joints 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, 7], the advantages of the present approach are the following: – 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 formulas, 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. The dynamics modeling of Gosselin and Gagné [33] is obtained using the Newton–Euler equations of dynamic equilibrium, which are written for each of seven moving rigid bodies of the three kinematical chains. The assumptions used here are that the joints connecting the distal links to the end-effector cannot transmit torques and consequently that they are equivalent to spherical joints. The Newton–Euler method, developed by Khalil in inverse dynamics of Orthoglide parallel robot [34] and Gough–Stewart platform [35], is based on applying the free-body diagram (FBD) procedure for each leg connecting the end-effector, where all components of joint forces and the actuated forces are unknown. Some interesting results are obtained by Tsai [36], applying the principle of virtual work. 2.2.2 Equations of Lagrange A solution of the dynamics problem of the 3-PRS 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 = x0G ,

q2 = y0G ,

q3 = z0G ,

q7 =

q8 =

q9 =

λA10 ,

A ϕ21 ,

λB10 ,

q4 = φ1 , q10 =

B ϕ21 ,

q5 = φ2 , q11 =

q6 = φ3 , λC10 ,

C q12 = ϕ21 .

(27)

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The Lagrange’s equations with their nine multipliers λ1 , λ2 , . . . , λ9 will be expressed by 12 differential relations, 9

∂L d ∂L = Qk + λs csk (k = 1, 2, . . . , 12) (28) − dt ∂ q˙k ∂qk s=1 which contain the following 12 generalized forces Q1 = 0, Q2 = 0, Q3 = 0, Q4 = 0, C A B Q5 = 0, Q6 = 0, Q7 = f10 , Q8 = 0, Q9 = f10 , Q10 = 0, Q11 = f10 , Q12 = 0, for example. A number of nine kinematical conditions of constraint are given by the relations (15): 12

csk q˙k = 0

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

(29)

k=1

The components of the general expression of the Lagrange function L = Lp + 2ν=1 (LAν + B C Lν + Lν ) are expressed as analytical functions of the generalized coordinates and their first derivatives with respect to time: 1 T 1 Lp = mp x˙0G2 + y˙0G2 + z˙ 0G2 + ω Jˆp ω p − mp gz0G , 2 2 p

i 1 iT i T Ci v10 − mi1 g uT3 r10 + p10 r1 , Li1 = mi1 v10 2 (30)

1 1 iT i iT i Ci i T i T Ci Li2 = mi2 v20 iT Jˆi ω v20 + ω i + mi2 v20 ω˜ 20 r2 − mi2 g uT3 r10 + p10 r21 + p20 r2 , 2 2 20 2 20 i = (A, B, C), 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 i i i i i i i v10 = λ˙ i10 u3 , v20 = p21 v10 , ω 10 = 0, ω 20 = ϕ˙ 21 u3 , ω˜ 20 = ϕ˙ 21 u˜ 3 , 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

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

=

(31)

T pk,k−1 u˜ 3 .

} (k = 1, 2, . . . , 12) of the A long calculation of the derivatives with respect to time dtd { ∂∂L q˙k 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 same expressions (23) C A B for the six input forces f10 , f10 , f10 required by the three prismatic actuators. As application let us consider a robot having the following architectural and mechanical characteristics: √ π π z0G∗ = 0.075 m, φ2∗ = , t = 3 s, r = 0.35 m, l = r 3, φ1∗ = , 36 36 h = 1 m,

λ0 = 0.35 m,

l1 = 0.95 m,

m1 = m2 = 0.5 kg, mp = 5 kg, ⎡ ⎤ ⎡ 0.3 4 2 ⎦ kg m , 0 Jˆ2 = ⎣ Jˆp = ⎣ 0.3

l2 =

(λ0 + l1 ) sin θ − r , cos(θ − β)

⎤ ⎦ kg m2 .

4 8

(32)


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2.3 Simulation procedure Based on the principle of virtual work, the procedure for solving the inverse dynamics of the 3-PRS spatial parallel robot can be summarized in several basic steps. 1. For a period of t = 3 second, it is assumed that the time-history evolution of the moving platform is specified in terms of its position and orientation about the center G from analytical equations (6). The relations (9), (10), and (11) give the evolution of joint variables i (i = A, B, C) and the parameters φ3 , x0G , y0G . λi10 , ϕ21 2. Using the relations (4) we compute the transformation matrices of three legs A, B, C : i i i i i , p21 and p20 = p21 p10 . p10 3. Determine the velocities and accelerations of all links by performing the inverse kinematics analysis in terms of prescribed velocities z˙ 0G , φ˙ 1 , φ˙ 2 and accelerations z¨0G , φ¨1 , φ¨2 of the moving platform. Specifically, for each leg, from the conditions of connectivity i i , ω21 and the relative accelerations (15) and (18) we compute the relative velocities v10 i i γ10 , ε21 . Cv Av Bv = 1, v10a = 0, v10a = 0, for example, 4. Using same equations (15), where we introduce v10a we compute the virtual characteristic velocity of each element of the robot. Other sets of Cv Bv = 1, v10c = 1. virtual velocities are obtained if we consider successively that v10b 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 . 7. For each link and platform we determine the inertia force Fkin (21) and the resulting force (excluding the actuator force) exerted to the rigid body Tk , from recursive equations (22). kin (21) and the 8. For each link and platform we determine the moment of inertia forces M resulting moment (excluding the actuator torque) exerted at the joint Ak , from recursive equations (22). i i A i and the power p10 = v10 f10 from the compact equation (23) 9. We find the input forces f10 established for the active prismatic joint. Using the MATLAB software, a computer program was developed to solve the inverse dynamics of the 3-PRS parallel manipulator. To illustrate the algorithm, it is assumed that for a period of three seconds the platform starts at rest from a central configuration and moves or rotates successively about two orthogonal directions. Assuming that there are no external forces and moments acting on the moving platform, a comparative study of the robots in two configurations: prismatic actuators (PRS) and revolute actuators (PRS) is based on the C C A B A B , p10 , p10 and p21 , p21 , p21 during computation of the power required by each actuator: p10 the evolution of the platform. The following examples are solved to illustrate the simulation. For the first example, the moving platform moves along the vertical z0 direction with variable acceleration while all the other positional parameters are held equal to zero. As can be seen from Fig. 3, it is proved to be true that all powers are permanently equal to one another in both actuation schemes. If the platform rotates about xG axis, the powers required by the actuators A1 , B1 , C1 A A or A2 , B2 , C2 are calculated by the program and plotted versus time as follows: p10 , p21 C C B B (Fig. 4), p10 , p21 (Fig. 5) and p10 , p21 (Fig. 6). Finally, the moving platform rotates about horizontal yG axis. From Figs. 7 and 8, we remark that the actuating powers of second and third actuators are equal. The simulation through the MATLAB program certify a permanent equality of the total power of the three actuators in the two configurations and that one of the major advantages of

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Fig. 3 Powers of three actuators

A , p A of first Fig. 4 Powers p10 21 actuator

the current matrix recursive formulation is a reduced number of additions or multiplications and consequently a smaller processing time of numerical computation in comparison with the methods based on the Lagrange formalism or Newton–Euler equations. Based on the Lagrange equations of second kind, the analytical inverse dynamics solved by Li and Xu [18] is expressed in terms of a set of three generalized coordinates since a general 3-PRS parallel mechanism has 3 DOF. However, this formalism will lead to a cumbersome expression for the Lagrange function, due to the complex kinematics of the manipulator. To simplify the analysis, using the principle of virtual work, the rotational inertias of the upper arms are neglected, assuming that the mass of each connecting rod is divided evenly and concentrated at its two extremities. Pursuing this simplifying hypothesis, a comparative study of two sets of powers C ∗C A B ∗A ∗B p10 , p10 , p10 and p10 , p10 , p10 , obtained in the present paper using the principle of vir-


253

B , p B of Fig. 5 Powers p10 21 second actuator

C , p C of third Fig. 6 Powers p10 21 actuator

tual work, is illustrated graphically for the vertical translation of the moving platform, for example, through the relative error-coefficient errpower =

A ∗A − p10 p10 × 100. A p10

(33)

From the simulation results (Fig. 9) we can see that is a deviation between the exact modeling and the simplifying procedure. The simplified analysis is the consequence of expressing of the geometrical constraint conditions in a reduced form, which are obtained from the remark that the distance between A2 and A3 is always equal to the length l2 of the intermediate connecting rod.

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Fig. 8 Powers of two actuators

3 Planar 3-RRR parallel robots Planar parallel robots are useful for manipulating an object on a plane. A mechanism is said to be a planar robot if all the moving links in the mechanism perform the planar motions. For a planar mechanism, the loci of all points in all links can be drawn conveniently on a plane. In a planar linkage, the axes of all revolute joints must be normal to the plane of motion, while the direction of translation of a prismatic joint must be parallel to the plane of motion. Aradyfio and Qiao [37] examined the inverse kinematics solution for the three different 3-DOF planar parallel robots. Gosselin and Angeles [38] and Pennock and Kassner [39] each present a kinematical study of a planar parallel robot, where a moving platform is connected to a fixed base by three links, each leg consisting of two binary links and three parallel revolute joints. Sefrioui and Gosselin [40] give an interesting numerical solution in the inverse and direct kinematics of this kind of planar robot.


255

Fig. 9 Relative error-coefficient of three actuators

Recently, Daniali et al. [41] have been presented a study of velocity relationships and singular conditions for a general planar parallel robot. Merlet [42] solved the forward pose kinematics problem for a broad class of planar parallel manipulators. Williams et al. [43] analyzed the dynamics and the control of a planar three-degree-of-freedom parallel manipulator at Ohio University, while Yang et al. [44] concentrate on the singularity analysis of a class of 3-RRR planar parallel robots developed in its laboratory. Bonev, Zlatanov and Gosselin [20] describe several types of singular configuration by studying the direct kinematics model of a 3-RPR planar parallel robot with actuated base joints. An interesting recursive inverse dynamics algorithm is validated by Khan, Krovi, Saha and Angeles in their papers [45, 46], for the specific case of parallel planar robots, based on the concept of the decoupled natural orthogonal complement and the Newton Euler equations. 3.1 Kinematics analysis Having a closed-loop structure, the planar parallel robot 3-RRR is a special symmetrical mechanism composed of three planar kinematical chains with identical topology, all connecting the fixed base to the moving platform. The centers A1 , B1 , C1 of three fixed pivots define the geometry of a fixed base and the three moving revolute joints A3 , B3 , C3 define the geometry of the moving platform. Each leg consists of two binary links with three parallel revolute joints. Together, the mechanism consists of seven moving links and nine revolute joints (Fig. 10). The planar parallel robot has two different actuation schemes. In a first kind of the robot (RRR) we consider the moving platform as the output link and the elements A1 A2 , B1 B2 , C1 C2 as the input links. Thus, all actuators can be installed on the fixed base. In the second configuration (RRR) three mobile actuators are fixed on the moving platform at the pivots centered in the points A3 , B3 , C3 . The origin of the Cartesian frame x0 y0 z0 (T0 ) is located at triangle center O, the z0 axis being perpendicular to the base and the x0 axis pointing along the direction C1 A1 . The origin of the mobile coordinate central system xG yG zG is located just at the center G of the moving triangle (Fig. 11). It is noted that the relative rotation of Tk body with ϕk,k−1 angle must be always pointing about the direction of zk axis.

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Fig. 10 The planar 3-RRR parallel robot

Fig. 11 Kinematical scheme of first leg A of the mechanism

In what follows we consider that the moving platform is initially located at a central configuration, where the platform is not rotated with respect to the fixed base, the mass center G is at the origin O of fixed frame, all legs are symmetrically extended and that the angles giving the initial pose of the mechanism have following values: αA =

π , 6

αB =

5π , 6

π αC = − . 2

(34)

One of three active legs (for example leg A) consists of a fixed revolute joint, a moving crank 1 of length l1 , mass m1 and tensor of inertia Jˆ1 , which has rotation about z1A axis with A A A A A , the angular velocity ω10 = ϕ˙ 10 and the angular acceleration ε10 = ϕ¨10 . A new the angle ϕ10 A A A element of the leg is a rigid rod 2 linked at the x2 y2 z2 frame, having a relative rotation with A A A A A , velocity ω21 = ϕ˙ 21 and acceleration ε21 = ϕ¨21 . It has the length l2 , mass m2 the angle ϕ21 ˆ and tensor of inertia J2 . Finally, a revolute joint is introduced at the moving platform, √ which is schematized as an equilateral triangle congruent to the base with the edge l = r 3, mass m3 and tensor of inertia Jˆ3 with respect to frame A3 x3A y3A z3A .


257

Starting from the reference origin O and pursuing the three legs OA1 A2 A3 , OB1 B2 B3 , OC1 C2 C3 we obtain nine transformation matrices [21] ϕ i p10 = p10 aα ,

pk0 =

k

ϕ p21 = p21 θ4 ,

pk−s+1,k−s

ϕ p32 = p32 θ4

(p = a, b, c), (i = A, B, C) (35)

(k = 1, 2, 3)

s=1

where

√ ⎡ −1 3 1 ⎣√ θ4 = 3 1 2 0 0

⎤ 0 0 ⎦. −2

(36)

In the inverse geometric problem, it can be considered that the position of the mechanism is completely given through the coordinates x0G , y0G of the mass center G of the moving platform and the orientation angle φ of the movable frame xG yG zG . We suppose that the position vector of G center r0G = [x0G y0G 0]T and the orientation angle φ, which are expressed by following analytical functions, can describe the general absolute motion of the moving platform in its vertical plane. π π π y0G = y0G∗ 1 − cos t , φ = φ ∗ 1 − cos t . (37) x0G = x0G∗ 1 − cos t , 3 3 3 The conditions concerning the absolute orientation of the moving platform are expressed by three identities ◦T p30 = R p30

(p = a, b, c),

(38)

where the resulting matrix p30 is obtained by multiplying three basic matrices p30 = p32 p21 p10 ,

R = rot(φ),

◦ p30 = p30 (t = 0) = aαi

(i = A, B, C).

(39)

From these conditions one obtains the first relations between the angles of rotation A A A B B B C C C − ϕ21 + ϕ32 = ϕ10 − ϕ21 + ϕ32 = ϕ10 − ϕ21 + ϕ32 = φ. ϕ10

The six variables equations, as follows: A + r10

C C A A B B , ϕ21 , ϕ10 , ϕ21 , ϕ10 , ϕ21 ϕ10

2

(40)

will be determined by several vector-loop

T A T GA B ak0 rk+1,k + a30 r3 = r10 +

k=1

2

T B T GB bk0 rk+1,k + b30 r3

k=1 C = r10 +

2

T C T GC ck0 rk+1,k + c30 r3 = r0G ,

(41)

k=1

where one denoted √ A = 0.5r[ 3 − 1 0]T , r10 i = r u1 , r21

i r32 = r u1 ,

B r10 = r[0 1 0]T ,

r3Gi = −r u1

√ C r10 = 0.5r[− 3 − 1 0]T , (i = A, B, C).

These vector equations give the inverse geometric solution for the manipulator: i i i i + αi − r sin ϕ10 + αi − ϕ21 − π/3 = y0G − y10 + r sin(φ + αi ), r sin ϕ10 i i i G i r cos ϕ10 + αi − r cos ϕ10 + αi − ϕ21 − π/3 = x0 − x10 + r cos(φ + αi ) (i = A, B, C).

(42)

(43)

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Supposing that the planar motion of the moving platform is known, we compute the linear and angular velocities of each leg in terms of the angular velocity ω 0G = φ˙ u3 and the center’s G velocity v0G = r˙ 0 of the moving platform. The following expressions give the velocities of the joints Ak and the angular velocities of compounding elements of leg A, for example: A A A A v10 = 0, v20 = ϕ˙ 10 a21 u˜ 3 r21 , A A A A A = a32 v20 + ϕ˙ 21 − ϕ˙ 10 , v30 a32 u˜ 3 r32 (44) A A A A A A A ω 20 = a21 ω 10 + ω 21 = ϕ˙ 21 − ϕ˙ 10 u3 , ω 10 = ϕ˙ 10 u3 , A A A A A A = a32 ω 20 +ω 32 = ϕ˙ 10 − ϕ˙ 21 + ϕ˙ 32 u3 . ω 30 Equations of geometrical constraints (40) and (41) can be derived with respect to time to obtain the matrix conditions of connectivity [47, 48]

A

A A T T T A T T GA A T T T GA ω10 uj a10 u˜ 3 r21 + a21 r32 + a21 a32 r3 uj a20 u˜ 3 r32 + a32 r3 + ω21 G A T T + ω32 uj a30 u˜ 3 rˆ3GA = uTj r˙ 0

(j = 1, 2)

(45)

A A A ˙ − ω21 + ω32 = φ. ω10 A A A , ω21 , ω32 as functions of anFrom these equations, we obtain the relative velocities ω10 G gular velocity φ˙ of the platform and the velocity r˙ 0 of mass center G and also the Jacobian matrix of the manipulator. Note that the matrix kinematical relations (44) and (45) will further be required in the computation of virtual velocity distribution of the elements of the manipulator. Assuming that the robot has successively two virtual motions determined by two sets of Cv Cv Av Bv Av Bv angular velocities ω10a =1, ω10a = 0, ω10a = 0 and ω32a = 1, ω32a = 0, ω32a = 0, the characteristic virtual velocities are expressed as functions of the pose of the mechanism by the general kinematical equations (45). These virtual velocities are required into the computation of virtual power and virtual work of all forces applied to the component elements of the robot. A A A As for the relative accelerations ε10 , ε21 , ε32 of the robot, the derivatives with respect to time of (45) give other following conditions of connectivity:

A

A A T T T A T T GA A T T T GA A T T ε10 uj a10 u˜ 3 r21 + a21 r32 + a21 a32 r3 uj a20 u˜ 3 r32 + a32 r3 uj a30 u˜ 3 rˆ3GA + ε21 + ε32

A G A A T T T A T T GA = uTj r¨ 0 − ω10 ω10 uj a10 u˜ 3 u˜ 3 r21 + a21 r32 + a21 a32 r3

A A A T T T GA A A T T − ω21 ω21 uj a20 u˜ 3 u˜ 3 r32 + a32 r3 ω32 uj a30 u˜ 3 u˜ 3 r3GA − ω32

A A A T T T T GA A A T T T T − 2ω10 ω21 uj a10 u˜ 3 a21 u˜ 3 r32 + a32 r3 ω32 uj a10 u˜ 3 a21 a32 u˜ 3 r3GA − 2ω10 A A T T T − 2ω21 ω32 uj a20 u˜ 3 a32 u˜ 3 r3GA

(46)

(j = 1, 2)

A A A ¨ − ε21 + ε32 = φ. ε10

If the other two kinematical chains of the robot are pursued, analogous relations can be easily obtained. The matrix relations (46) will be further used for the computation of wrench of the inertia forces for every rigid of the robot. 3.2 Dynamics equations Knowing the position and the kinematics state of each link as well as the external forces acting on the robot, this paper derives the final form of explicit dynamics equations of the


259

3-RRR planar constrained robotic system, using the principle of virtual work. Three electric A10 = mA10 u3 , m B10 = mB10 u3 , m C10 = mC10 u3 motors A1 , B1 , C1 that generate the moments m oriented about fixed parallel axes or another three mobile motors acting with the moments B32 = mB32 u3 , m C32 = mC32 u3 control successively the motion of the moving m A32 = mA32 u3 , m platform. 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 moving platform, and takes its effect into account by introducing the corresponding constraint conditions. The first and more complicated open tree system includes the acting link and could comprise the moving platform. Considering three independent virtual motions of the robot, the virtual velocities should be compatible with the virtual motions imposed by all kinematical constraints and joints at a given instant in time. Using the Jacobian matrix expressed by the conditions of connectivity v v ,ω k0 associated with all moving links are related to a (45), the absolute virtual velocities vk0 v v = ωk,k−1 u3 . set of independent relative virtual velocities ω k,k−1 Assuming as negligible the frictional forces at the joints, the virtual work produced by all forces of constraint at the joints is zero. Total virtual work contributed by the first actuator kin and by the torque m A10 , for example, inertia forces and moments of inertia forces Fkin , M ∗ ∗ wrench of known external forces Fk , Mk can be written in a compact form, based on the v . Applying the fundamental principle of virtual relative virtual angular velocities only ωk,k−1 work the following compact matrix relations results:

A Av A Av A Bv B Cv C 1 + ω21a M2 + ω32a M3 + ω21a M2 + ω21a M2 (47) mA10 = uT3 M for the torque of first fixed revolute actuator and

Av A Av A Bv B Bv B Cv C Cv C 3A + ω10a 1 + ω21 M M2 + M M1 + ω21 M2 + ω10a M1 + ω21 M2 (48) mA32 = uT3 ω10a for the torque of first mobile revolute actuator. The relations (47) and (48) represent the inverse dynamics model of the 3-RRR planar parallel robot. As an application let us consider a 3-RRR planar robot, which has the following characteristics: √ π r = 0.3 m, l = r 3, y0G∗ = 0.025 m, φ∗ = , x0G∗ = −0.025 m, 12 (49) l1 = l2 = 0.3 m, t = 3 s, m1 = 3 kg, m2 = 1.5 kg, m3 = 5 kg. To illustrate the algorithm, it is assumed that for a period of three second the platform starts at rest from a central configuration and rotates or moves along rectilinear directions. A comparative analysis of the manipulators in two configurations: fixed revolute actuators and mobile revolute actuators is based on the computation of the power required by C C A B A B , p10 , p10 , p32 , p32 , p32 during the platform’s evolution. each actuator: p10 For the first example we consider the rotation motion of the moving platform about z0 axis with variable angular acceleration while all the other positional parameters are held equal to zero: Figs. 12, 13, and 14. If the platform’s center G moves along a rectilinear planar trajectory without rotation of platform, the powers required by the actuators A1 , B1 , C1 or A3 , B3 , C3 are calculated by the program and plotted versus time as follows: Figs. 15, 16, and 17. C C B B , p32 (Fig. 13) or p10 , p32 (Fig. 14), it is remarked that the secAs can be seen from p10 ond mobile actuator and the third fixed actuator during the rotation motion of the platform are working in an active regime with comparable values of the absorbed powers. The total

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tot tot powers p10 , p32 developed by the complex of three actuators in the case of the two driving configurations during the period of the same general evolution of the moving platform, are identically distributed both for the rotation motion (Fig. 18) and the rectilinear translation of the moving platform (Fig. 19). As far as the motion of the platform is decelerated during the second part of the time interval t , the power graph has an anti-symmetrical configuration in the hypothesis of a non-uniform rotation performed around the mass center. The simulation program certify that one of major advantages of the current matrix recursive approach is the well structured way to formulate a dynamic model, which leads to a computational efficiency. The proposed method can be applied to various types of complex robot, when the number of components of the mechanism is increased.


261

C , p C of Fig. 14 Powers p10 32 third actuator


4 Conclusions Most of 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 this approach is equal to the total number of the position variables and Lagrange multipliers inclusive. Also, the analytical calculations involved in these equations present a risk of errors. 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 connecting forces in the joints. Finally, the actuating torques could be obtained. Within the inverse kinematics analysis some exact relations that give in real time the position, velocity and acceleration of each element of the parallel robot have been established in the present paper.

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C , p C of Fig. 17 Powers p10 32 third actuator

Based on the principle of virtual work, this approach can eliminate all forces of internal joints and establishes a direct determination of the time-history evolution of powers required by the actuators. The dynamics model takes into consideration the mass, the tensor of inertia and the action of weight and inertia force introduced by all compounding elements of the parallel mechanism. The methodology can be available for the kinematics analysis and the nonlinear dynamics of a multibody systems consisting of interconnected rigid and, eventually, deformable bodies, each of which may undergo large translational and rotational displacements. Examples of mechanical structures that can be modeled as constrained robotic systems are: mechanisms, machines, parallel robots, mobile robots, gear trains for robotics, and spatial robotic hybrid structures. Choosing appropriate serial kinematical circuits connecting many moving platforms, the present method can easily be applied in forward and inverse mechanics of various types of


263

tot , p tot Fig. 18 Total powers p10 32 of three actuators

tot , p tot Fig. 19 Total powers p10 32 of three actuators

parallel mechanism, complex manipulators of higher degrees of freedom, when the number of components of the mechanisms is increased. Concerning the power requirement comparison, we remark that for the same evolution of the moving platform, the power distribution upon the three actuators depends on the actuating configuration, but the total power absorbed by the set of three actuators is the same, at any instant, for both driving systems.

References 1. Tsai, L.-W.: Robot Analysis: The Mechanics of Serial and Parallel Manipulators. Wiley, New York (1999) 2. Stewart, D.: A platform with six degrees of freedom. Proc. Inst. Mech. Eng., Part 1 180(15), 371–386 (1965) 3. Merlet, J.-P.: Parallel Robots. Kluwer Academic, Dordrecht (2000)

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