KINEMATIC MODELLING AND SIMULATION OF A NOVEL

@inproceedings{BMZ_MIC04, author="L. Bruzzone and R.M. Molfino and M. Zoppi", title="Kinematic modelling and simulation of a novel interconnected-chains PKM", booktitle="Int. Conf. Modelling, Identification and Control MIC2004", month="February 23-25", address="Grindelwald, Switzerland", isbn="0-88986-387-3", issn="1025-8973", year="2004" }

KINEMATIC MODELLING AND SIMULATION OF A NOVEL INTERCONNECTED-CHAINS PKM L. Bruzzone, R. Molfino, M. Zoppi PMAR Robot Design Research Group – University of Genova Via All’Opera Pia 15A, 16145 Genova Italy

Abstract The paper deals with the conception and the kinematic modelling of a 5-degree-of-freedom InterconnectedChains Parallel Kinematics Machine (IC-PKM). Usually parallel kinematics machines are composed of a platform connected in parallel to a fixed base by independent actuated serial chains. For this reason, the inverse position analysis is relatively trivial, while the direct position analysis is much more difficult. In the proposed machine, the serial chains, which connect the end-effector to the fixed base, are not independent. For this reason, both the direct and the inverse position analysis are represented by coupled algebraic systems, which are discussed in the present paper. A multibody model has been realized in order to assess the dynamic performance of the system.

The second approach has been used to synthesize a machine capable of performing the motion represented in Fig. 1. The end-effector, represented by the coordinate frame C, moves on the surface of a sphere (3 degrees of freedom, defined by the three spherical angles ϕ, θ, ψ); moreover, the radius of the sphere (r) is variable and the sphere center O moves along a base axis (z); in total, the end-effector position and orientation are defined by the five external coordinates z, r, ϕ, θ, ψ (Fig. 1). The industrial usefulness of a machine characterized by this kind of mobility is remarkable, because it can be applied to machining, finishing and cleaning of spherical surfaces, such as lenses. Therefore our research has been focussed on the synthesis of a parallel robot capable of performing the previously described mobility.

Key Words Parallel Kinematics Machine, Position Analysis, Dynamic Simulation.

1. Introduction Nowadays parallel kinematics machines (PKM) represent one of the most interesting research fields in robotics. Their peculiar features, in particular the high dynamic performance due to the low moving masses and the great structural stiffness due to the closed-loop chains, can be exploited in many applications that are critical for traditional serial manipulators [1]. On the other hand, a great drawback is related to the smallness of the workspace in comparison with the overall dimensions; moreover, the dexterous workspace (with sufficiently wide orientation capability) is usually a small part of the reachable workspace [2]. For this reason, the design of a parallel kinematic machine is, in most cases, application-oriented [3]. There are two possible approaches to design a PKM that fulfils specific task requirements: the first is to select a known mechanical architecture and to evaluate suitable geometrical parameters; the second is to conceive a completely novel architecture on the basis of the task geometry [4][5].

Fig. 1 – external coordinates defining the considered mobility

Fig. 2 – the proposed parallel kinematics machine

2. The Proposed Parallel Kinematics Machine Fig. 2 shows the kinematic model of the proposed machine. The position and orientation of the end-effector {1} are represented by its central point C and by the coincident reference frame. The end-effector is connected by revolute joints to three equal beams {2}. The axes of the revolute joints are coincident, and represent the endeffector symmetry axis z. At the end of each beam there is another short beam, perpendicular to the previous one and rigidly connected to it. The latter supports two spherical joints, symmetrically placed with respect to the plane containing the first beam and the revolute joint axes. Keeping the end-effector fixed, all the six spherical joint centers can rotate due to the revolute joints, always remaining in the same plane. The fixed base is represented by the equilateral triangle G1G2G3; the fixed reference frame is located in the centroid G of G1G2G3. The fixed base is connected by prismatic joints to three translating sliders {4}, which are placed with 120° symmetry: each prismatic joint axis passes through one of the points G1, G2 and G3 and lies in a vertical plane that contains the point G. The three prismatic joints are actuated. Each slider carries two universal joints, symmetrically placed with respect to the plane that contains the corresponding prismatic joint axis and the machine symmetry axis. Six equal beams {3} are connected to one of the sliders {4} by a universal joint at one end, and to one of the beams {2} by a spherical joint at the other end. If two spherical joints were used to connect each beam {3}, a free rotation about its axis would be possible.

The end-effector is connected to the base also by another kinematic chain: the beam {6} is connected to the base by a cylindrical joint, with two actuated degrees of freedom, one translational and one rotational, along the same axis; this axis corresponds to the machine symmetry axis (intersection of the three machine symmetry planes). The end-effector is connected to the beam {6} by the extensible leg {5}, endowed with a passive prismatic joint with axis coincident to the end-effector symmetry axis; one end of {5} is rigidly connected to the end-effector; {5} and {6} are connected by an universal joint placed in O, which is the center of the spherical motion of the endeffector: in fact, due to the kinematic bounds related to the members {5} and {6}, the allowable motion is evidently the one shown in Fig. 1. The motion of the end-effector is driven by five actuators, corresponding to five internal coordinates (q1,…,q5): three actuate the sliders {4} (q1, q2, q3) and two actuate the two degrees of freedom of the beam {6} (q4: translation with respect to the fixed frame; q5: rotation with respect to the fixed frame). The two spherical joints connected to each one of the three beams {2} can be coincident; in this case the kinematic equations can be simplified using a geometrical approach [6]. In the following of this paper the general case will be treated, with non-coincident spherical joints. A constructive model of the robot is shown in Fig. 3.

3. Position Analysis: Preliminaries The rotation of the end-effector about its symmetry axis is represented by the external coordinate ψ. It is evident that, considering known the other external coordinates, ψ is a function of the internal coordinate q5; this function is

easily obtainable by the kinematic analysis of universal joint in O, and depends on the orientation of the universal joint axes. On the other hand, in the inverse kinematic analysis the angle ψ has no influence on the internal coordinates q1 to q4, but only on q5. For these reasons, in the direct kinematic analysis ψ can be determined after the other external coordinates, which are not influenced by q5. In a similar way, in the inverse kinematic analysis q5 can be determined after the other internal coordinates, which are not influenced by ψ.

where the function arctan2(y,x) calculates the arc tangent of y/x using the signs of both arguments to determine the quadrant of the result. Considering fixed the position of C, the three beams {2} connected to the end-effector can rotate in the plane perpendicular to r and passing through C: the position of the i-th beam {2} is defined by the angle βi from the plane uuuur containing the points O, C, G to the vector CCi (Fig. 4).

Fig. 4 – definition of the angle βi The unit vector with direction from O to C is obtainable by means of the following rotation matrix product, which expresses the ϕ and θ rotations of the base z-axis: v OC = R z ( ϕ ) R y ( θ ) [ 0 0 1] = [ cϕsθ sϕsθ cθ] T

Fig. 3 – constructive model of the robot Therefore, in order to simplify the treatment, in the position analysis we consider only the core of the problem, which is the direct and inverse relation between the internal coordinates q1 to q4 and the external coordinates z, r, ϕ, θ; to this aim, in the following of the paper we will use the four-dimensional vectors q = [ q1

q2

q3

q4 ]

T

and x = [ z r ϕ θ] . Let us T

note that the internal coordinate q4 coincides with the external coordinate z. The direct and inverse relation between the vector r from O to C and the external coordinates r, ϕ, θ are the following: rx = r sin θ cos ϕ   ry = r sin θ sin ϕ  r = r cos θ  z

 r = rx2 + ry2 + rz2   ϕ = arctan2 ( ry ,rx )   θ = arccos rz rx2 + ry2 + rz2 

(

(1)

(3) where the abbreviations s = sin and c = cos are used, as in the following. The unit vector of the direction of the i-th beam (from C to Ci) is calculated considering the ϕ , θ and βi rotations of the base x-axis: v i = R z ( ϕ ) R y ( θ ) R z ( βi ) [1 0 0] = T

= [ cϕcθcβi − sϕsβi

)

sϕcθcβi + cϕsβi

(4)

− sθcβi ]

T

while the same rotations applied to the base y-axis provides the unit vector of the direction from Ci to one of the connected spherical joints: w i = R z ( ϕ ) R y ( θ ) R z ( βi ) [ 0 1 0 ] = T

= [ −cϕcθsβi − sϕcβi

(2)

T

− sϕcθsβi + cϕcβi

sθsβi ]

T

(5)

The positions of the six spherical joint centers Bi (Fig. 2) can be expressed as functions of the components of x and of the three angles βi: b B1 = zk + r ( r, ϕ, θ ) + dv1 ( ϕ,θ,β1 ) + w1 ( ϕ,θ ,β1 ) 2

Eqs. (8) represent a system of six non-linear algebraic equations in the six unknowns r, ϕ, θ, β1, β2, β3. No explicit solution of this system has been found, that has been solved numerically imposing proper inequalities to avoid solutions without physical meaning.

b B 2 = zk + r ( r, ϕ, θ ) + dv 2 ( ϕ ,θ ,β 2 ) − w 2 ( ϕ,θ ,β2 ) 2 b B3 = zk + r ( r, ϕ, θ ) + dv 2 ( ϕ,θ,β 2 ) + w 2 ( ϕ,θ ,β2 ) 2 b B 4 = zk + r ( r, ϕ, θ ) + dv 3 ( ϕ ,θ,β3 ) − w 3 ( ϕ,θ ,β3 ) 2 b B5 = zk + r ( r, ϕ, θ ) + dv 3 ( ϕ,θ,β3 ) + w 3 ( ϕ,θ ,β3 ) 2 b B 6 = zk + r ( r, ϕ, θ ) + dv1 ( ϕ ,θ ,β1 ) − w1 ( ϕ,θ ,β1 ) 2

5. Inverse Position Analysis

(6) where: k is the unit vector of z-axis of the base reference uuuur frame; d is the length of the three vectors CCi ; b is the uuuur uuuuur uuuuur length of the three vectors B6 B1 , B2 B3 , B4 B5 . The positions of the six universal joint centers Ai (Fig. 2) can be expressed as functions of the internal coordinates q1, q2, q3: T

 3 l a sαq1 + − A1 =  2 4  2

 sα 3 3 − q1 − l− a cαq1  2 6 4 

 3 l a sαq1 + + A2 =  2 4  2

 sα 3 3 − q1 − l+ a cαq1  2 6 4 

a A3 =  2

 cαq2  

3 l + sαq2 3

 a A 4 = −  2

T

 cαq2  

3 l + sαq2 3

T

T

 3 l a sαq3 − − A5 =  − 2 4  2

 sα 3 3 − q3 − l+ a cαq3  2 6 4 

T

 3 l a A6 = − sαq3 − + 2 4  2

 sα 3 3 − q3 − l− a cαq3  2 6 4 

T

(7) where: α is the angle between the base z-axis and the each prismatic joint axis, positive if the prismatic guide points outward (Fig. 2); l is the side of the equilateral triangle uuuur uuuuur G1G2G3; a is the length of the three vectors A1 A2 , A3 A4 , uuuuur A5 A6 . Obviously, each position vector is function of only one internal coordinate.

4. Direct Position Analysis If the vector q is known, the six position vectors Ai can be obtained by means of eqs. (7). As we have already said, z = q4. It is possible to impose that the distance between Ai and Bi is equal to the length m of the six beams {3}:

( A ( q ) − B ( x , β ,β i

i

1

,β3 ) ) = m , i = 1…6 2

2

2

(8)

If the vector x is known, the six position vectors Bi can be expressed as functions of β1, β2, β3 by means of (6); moreover, q4 = z. It is possible to use eqs. (8) also in the inverse kinematic analysis: they represent a system of six non-linear algebraic equations in the six unknowns q1, q2, q3, β1, β2, β3. No explicit solution of this system has been found, that has been solved numerically imposing proper inequalities to avoid the solutions without physical meaning. The kinematic chains from the fixed base to the endeffector are interconnected (the machine belongs to the class of the Interconnected-Chains PKM, IC-PKM): each beam {2} is connected to two sliders {4}, and each of these sliders is connected to another beam {2}. For this reason, it is impossible to solve the inverse position analysis of a IC-PKM considering separately each kinematic chain that drives the end-effector; on the contrary, this is possible for the parallel robots which belong to the class of HCM [7]. This results in a complex, fully coupled inverse position analysis, differently from most parallel kinematics machines.

6. Geometrical Approach to the Inverse Kinematic Analysis The inverse position problem can be analyzed, with noncoincident or coincident spherical joint pairs, by using a geometrical approach instead of solving system (8). In the inverse position problem, the external coordinates are known. Therefore, each position vector Bi, according to (6), is a function of only one of the three angles β1, β2, β3. Each point Ai can be determined as one of the two intersections between: -

the sphere Si with center Bi (βj) and radius m;

-

the straight line pi that is the locus of the possible positions of Ai due to the corresponding prismatic joint.

For constructive reasons, only the solution characterized by a lower value of the coordinate z Ai is significant and has to be considered. It is possible to impose the following conditions: z A1 ( β1 ) = z A2 ( β 2 ) z A3 ( β 2 ) = z A4 ( β 3 ) z A5 ( β 3 ) = z A6 ( β1 )

(9)

Eqs. (9) represent a system of three non-linear algebraic equations in the three unknowns β1, β2 and β3. Once obtained these unknowns, the center Bi of Si can be computed by (6), then the intersection between Si and line pi allows to calculate qi. In the inverse position problem, system (9) replaces the system of six non-linear algebraic equations in six unknowns described in the previous section. Due to the lower number of unknowns, the numerical solution of this system is simpler.

8. Kinematic and Dynamic Simulation A multibody model has been built using the software package ADAMS (Fig. 5) in order to asses the dynamic performance of the machine. In the following some simulation results are shown, with reference to: - a model characterized by the geometric and inertial parameters reported in Tab. 1 (the beams are considered with constant section); - a circular trajectory with constant speed characterized by the parameters reported in Tab. 2 (let us note that the rotation senses of ϕ and ψ are opposite). l

α m a b d mass of the end-effector {1} mass of one beam {2} mass of one beam {3} mass of one translating slider {4} mass of the variable-length beam {5} mass of the beam {6}

0.5 m 0 rad 0.55 m 0.1 m 0.06 m 0.15 m 1 kg 1.5 kg 1.25 kg 5 kg 2.5 kg 2.0 kg

Tab. 1 – geometrical and inertial parameters

Fig. 5 – multibody model of the mechanism

z r ϕ θ ψ

0.2 m 0.3536 m π/2 rad - 2π rad/s · (time) 0.142 rad −π/2 rad - 4π rad/s · (time)

Tab. 2 – circular trajectory: external coordinates timehistories

7. Jacobian Matrix The Jacobian matrix of the robot has been obtained applying the screw theory; the equations are reported in [6]. The rank of the Jacobian matrix has been numerically assessed in order to check the correct mobility of the machine: since the matrix is full-rank out of singularities, the end-effector motion can be controlled by the actuators.

The machine is considered frictionless, and the gravity is along the z-axis. In fig. 6 the time histories of the internal coordinates q1, q2, q3 and q5 during a complete turn are shown (q4 is constantly equal to 0.2 m). Since the angle ψ remains constantly null according to the reference shown in Fig. 1 the coordinate q5 performs a 360° rotation.

Fig. 6 – simulation results: internal coordinates

Fig. 7 – simulation results: time derivatives of the internal coordinates

Fig. 8 – simulation results: actuation forces and torques In Fig. 7 the time derivatives of the internal coordinates are shown. As regards the dynamic results, Fig. 8 shows the actuation forces F1 to F4 related to the coordinates q1 to q4 and the actuation torque τ5 related to the coordinate q5.

9. Conclusions A novel 5-degree-of-freedom Interconnected-Chains PKM has been designed considering the geometry of a particular class of tasks, characterized by a spherical motion of the end-effector. The direct and inverse kinematic problems have been analysed. The end-effector is not connected to the fixed base by independent serial chains, as occurs for most parallel kinematics machines: since the kinematic chains that drive the end-effector are interconnected, not only the direct, but also the inverse position problem is represented by a coupled non-linear algebraic problem, while usually for parallel kinematics machine in the inverse problem each internal coordinate can be obtained independently. However, applying a geometrical approach, the complexity of the problem has been reduced (from six equations in six unknowns to three equations in three unknowns). The synthesis of application-oriented IC-PKM seems to be an interesting solution for a wide class of applications requiring high accuracy. As a matter of fact, the structural stiffness of an interconnected closed-loop chain is usually higher than the one of a parallel closed-loop chain.

Therefore, the theoretical aspects related to this kind of machines seem to be an interesting research field.

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