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DESIGN OF COMPLIANT PARALLEL KINEMATIC MACHINES

Yong-Mo Moon Research Fellow

Mechanical Engineering University of Michigan Ann Arbor, MI 48109

ymmoon@engin.umich.edu

Sridhar Kota Professor

Mechanical Engineering University of Michigan Ann Arbor, MI 48109

kota@umich.edu

ABSTRACT Conventional Parallel Kinematic Machines (PKMs) suffer

from poor repeatability due to backlash in various joints. Use of flexures to overcome this limitation has also been reported in the published literature. However, conventional flexures suffer from axis drift, limited range of motion and stress concentrations. We present a new method of design of Compliant PKMs (CPKMs) employing a new type of compliant joints. We employ large displacement compliant joints to construct PKMs, which exhibit sub-micron accuracy. The methodology presented in this paper allows us to synthesize a PKM for any desired number and type of kinematic degrees of freedom by employing modular construction elements. A library of constraining legs synthesized from compliant joints is used for construction of any CPKM. The paper presents a systematic approach to transforming an arbitrary 3D motions specification into a set of motions along orthogonal axes using dual vector representation. The method enables construction of any CPKM by using the library of standard constraining legs and active legs. The methodology is illustrated with a design example.

INTRODUCTION

A typical parallel kinematic machine (PKM) is comprised of a moveable platform (top plate) and a fixed base (bottom plate) which are connected by a set of legs. Each connects to the platform and the base with a set of mechanical joints (revolute, universal, prismatic or spherical). Active legs are the ones which have an actuator (typically a linear actuator) and the passive ones do not. Depending on the desired number and the type of kinematic degrees of freedom of the platform, in the current practice, we establish the number and spatial orientation of the legs, the number and the type of actuators and the type of joints. The original Gough-Stewart platform design was introduced for aircraft simulator with six degrees of freedom

[Stewart 1965]. Numerous variations of the “Stewart’s Platform” have been reported in the literature with varying degrees of freedom (dof), joints and actuation schemes. A majority of designs that are reported in the literature are either 6-dof spatial mechanisms or 3-dof planar mechanisms. Many studies have shown that PKMs are inherently stiffer than conventional serial robots. Three-dimensional positioning and manipulation mechanisms with high precision motion are needed in many applications such as manufacturing equipment and MEMS devices [Dasgupta 2000]. As with any mechanical system composed of joints, PKMs suffer from errors due to backlash, hysteresis and manufacturing errors in the joints. Therefore it has always been a major challenge to achieve high precision with conventional joints. In order to overcome this limitation, designers have successfully employed flexures for positioning and manipulation applications which demand very high precision [Hara & Sugimoto 1989, Prenette and etc. 1997]. Many, if not all, of such flexure designs typically employ a notch-type flexure which suffer from high stress concentrations and are therefore limited to very small range of motions.

In this paper, we will first briefly describe complaint joints

which we later employ in compliant PKMs. We will then describe a new methodology for synthesis of CPKMs. This generic method applies to conventional as well as compliant PKMs.

COMPLIANT JOINTS We introduce three basic types of complaint joints: (1)

compliant revolute (CR) joint, (2) compliant universal (CU) joint and (3) compliant translational (CT) joint. The compliant joints reported here are different from flexures in that the compliant joints:

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(i). Utilize distributed compliance of beams to provide joint flexion

(ii). Have better stress distribution (lower stress concentrations) (iii). Provide greater range of relative motion (iv). Motion axis does not drift as the joint is flexed.

Detail discussion on complaint joint design and

comparisons to other type of flexures is given in a companion paper titled “ Large Displacement Compliant Joints” and is therefore not repeated here.

Compliant Revolute Joint The CR joint is designed to provide very large compliance

and relatively large range of motion compared to the notch type revolute joints. The cross section of the CR joint has ribs to prevent bending and reduced cross-sections to provide torsional compliance. Other variations of this joint including design details are given in a companion paper [Moon, Trease, and Kota 2002]. The axis of rotation drifts minimally compared to conventional flexures. For instance, a joint with beam thickness of 1 mm, overall width of 10 mm, beam length of 50mm, the axis drifts only 215 nanometers when a force of 1 Newton is applied in the lateral direction. The lateral stiffness of CR joints shown below range from 33 to 500 times the joint stiffness.

(a) Center-Moment CR joint (b) End-Moment CR joint

Figure 1. Cross Type Compliant Revolute joints

Compliant Universal Joint The compliant universal (CU) joint is simply a combination

of two CR joints. The rotational axes of the joints intersect at the center of the joint.

Figure 2. Compliant Universal Joint

Compliant Translational Joint The planar compliant translational (CT) joint employs a set

of four parallelograms in an over-constrained arrangement of parallel beams (leaf springs). The joint allows a large range of motion since the load is shared by more number of relatively thin beams. Straight-line motion is achieved by the symmetry about the longitudinal axis.

(a) Spatial Prismatic Joint (b) Planar Prismatic Joint

Figure 3. Compliant Prismatic Joints The planar CT joint has two sets of six parallel cantilever

beams (leaves) connected in series. The lateral or off-axis stiffness of the planar CT joint is at least 60 times the axial stiffness (compliance in the direction of motion). The lateral stiffness of the spatial CT joint is over 80 times its axial stiffness. Readers are encouraged to refer to our companion paper [Moon, Trease, Kota 2002 ] for analytical expressions and design charts for sizing CR and CT joints for a given application.

GENERAL CONFIGURATION OF COMPLIANT PARALLEL KINEMATIC MACHINES

Figure 4(a) shows a six-degree of freedom PKM

constructed from compliant joints. Each leg has six degrees of freedom through a series of compliant joints; a ball joint (combination of a CU and a CR joint) connection with the platform, a translational joint and a CU joint connection with the base. None of the legs is active. The top platform can rotate about and translate along X, Y and Z-axes. To actively control the position of the platform, we replace the passive CT joints with linear actuators (Figure 4b). The number of active legs (and hence the number of actuators) is equal to the total number of desired degrees of freedom of the platform. However, if fewer than six degrees of freedom are desired, we simply add what is called a constraining leg with the same number and type of degrees of freedom as desired for the top platform. The constraining leg typically connects the center of the top platform (bottom surface) with the center of the base (see figure 6). If a six dof constraining leg, or serial kinematic chain of joints with six degrees of freedom (figure 5(a)), is incorporated into a six dof PKM shown in figure 4, it would not affect the total dof of the platform. However, if a four dof serial kinematic chain (figure 5(b)) is added to the six dof PKM of figure 4 as a constraining leg, then the platform would have only the same

Motion Axis

Link Axis Motion Axis

Link Axis

X

Y

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four degrees of freedom as that of the constraining leg. That is, the platform is constrained to the same four kinematic degrees of freedom as the constraining leg. In which case, we need only four active legs each with a linear actuator, for a 4 dof PKM. Note that each active leg must allow six degrees of freedom. Typically, the ends of the active legs are connected to the platform and base by universal joints (CU joints) as shown in Figure 5 (c).

(a) Passive 6-dof CPKM (b) Active 6-dof CPKM

Figure 4. Six Degree of freedom CPKMs

(a) Six dof Serial Kinematic Chain

(b) A four dof constraining leg

(c) A generic six dof Active Leg of a CPKM with two universal joints at each end and a linear actuator (cylindrical joint)

Figure 5: Constraining Legs and Active Legs of CPKM

Figure 6 shows a general configuration of a CPKM. The trace of the origin (O’) of the platform defines the desired workspace. Active leg is connected to the platform and the base at points A and B with CU joints.

B

A

X

Y

Z

X’ Y’

Z’

O

O’

ConstrainingLeg

Active LegR

Figure 6. Schematic representation of CPKM

This method can be generalized to configuration of PKMs

with any number of degrees of freedom. Depending on the number of degrees of freedom one can select from a library of constraining legs shown in Table 1. For instance to configure a three degree of freedom PKM, we simply select an appropriate constraining leg from Table 1 and incorporate three active legs as shown in Figure 7. Thus, establishing the configuration (type synthesis) of a PKM is relatively straightforward. This simple design procedure has several advantages. First, the structure of the active legs is generic. The number of active legs to be employed is equal to the number of degrees of freedom of the PKM. If the desired motion of the platform is specified as a set of rotations and translations along X, Y and Z orthogonal axes, then Table 1 can be used to select an appropriate constraining leg depending on the number and types of degrees of freedom. Once this selection is made, the stiffness of each the compliant joints must be optimized according to the range of motion and desired off-axis stiffness [Moon, Trease, and Kota 2002]. However, if the platform is required to move in an arbitrary 3d space, then the desired motion must be decomposed into a set of motions (rotations and translations) along orthogonal axes which duplicate the desired motion. In the next section we present a systematic method of decomposing arbitrary motions into a set of orthogonal motions using dual vector representation.

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Figure 7. A 3 dof CPKM comprising of a 3dof constraining leg and 3 standard active legs.

MOTION REPRESENTATION IN DUAL VECTORS The desired motions in 3D space are represented by dual vectors [Moon 2000]. Dual vector captures three components of motion: range of motion (magnitude), type of motion (pitch), and position & orientation (line) of the motion. These quantities are represented as dual vector in the form: S = [M] (PA + εPL) {l + εl0} (4) where, AP and LP are angular and linear pitches respectively, and [ ]M is magnitude. The definitions are,

[ ][ ][ ]

≠=

length otherwise 0P if angle isunit Thediscrete ismotion theif : M;,M,Mcontinuous ismotion theif :M;MM

M

A

current10

currentmaxmin

Λ ,

l represents the direction of the motion and l0 represents the origin of the motion axis.

Once the required motion is represented by dual vectors,

the pitch of the dual vector is used to determine the type of compliant joint to be used. It also indicates if the motion needs to be decomposed into rotations and translations along orthogonal axes. The magnitude of the dual vector is determining proper stiffness of (sizing) corresponding compliant joints. The line vector is used to determine the placement (position and orientation) of corresponding compliant joint.

The dual vector representation of a revolute joint (CR joint) and translational joint (CT joint) respectively are:

[ ]( ){ }0εε01M ll ++ and [ ]( ){ }0εε10M ll ++ . (5)

Each compliant joint has its own screw in dual vector form

and the dual vectors of the required motion can be mapped to types of compliant joint by matching angular and/or linear pitch. However, the compliant joints in this paper are designed for only pure rotational and pure translational motions. Therefore, the pitch can be either (1+ε0) or (0+ε1). Any other

pitch suggests that the dual vector should be decomposed further into a set of pure rotational and/or pure translational pitch (joints). The ordered set of decomposed dual vectors suggests an ordered sequence of compliant joints that make up the constraining leg.

TYPE SYNTHESIS OF CONSTRAINING LEG We describe two different schemes for type synthesis of

constraining leg: (i) when desired motion is expressed in terms of kinematic degrees of freedom along x, y and z axes, and (ii) when the desired motion is expressed as a free form motion in three dimensional space.

Case 1: Type Synthesis for Given Degrees of Freedom When the required motions are given in form of number

and type of degrees of freedom about x, y, z axes, the constraining leg design involves assembly of appropriate compliant joints according to the prescribed dofs. Therefore, the design process of the constraining leg ends at the type synthesis for these cases. Depending on the number and type of dof, one can select from the library of constraining legs shown in Table 1. For instance to configure a three degree of freedom PKM, we simply select an appropriate constraining leg from Table 1 and incorporate three active legs as shown in Figure 7.

Thus, establishing the configuration (type synthesis) of a

PKM is relatively straight forward. This simple design procedure has several advantages. First, the structure of the active legs is generic. The number of active legs to be employed is equal to the number of dof of the PKM. Depending on the number and the type of dof desired, the user could readily select an appropriate constraining leg from Table 1. The stiffness of the compliant joints must be optimized for a given range of motion. The details of compliant joint design are described in a companion paper [Moon, Trease and Kota 2002 ].

Case 2: Type Synthesis for Analytic Motion Requirements

Here the motion requirements are given in form of functions or geometric entities, and therefore needs proper interpretation and decomposition into elemental motions along x, y, and z axes. For example, suppose an end-effector is required follow three-dimensional contours of a part as shown in figure 8. For simplicity sake, we further assume that the contour is a section of an elliptical ring (figure 8) The required motion in dual vector form (equation 6) shows that the elliptical ring surface can be decomposed into a circular section and a translation. The required degrees of freedom are three since the motion along the elliptical ring require two degrees of freedom and a third degree of freedom to move along the perpendicular direction. In dual vector form,

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( )[ ]LaLT ×++= ε)sin(d ε θθ (6)

where, eccentricities of the ellipse d and direction vector of L are variables.

Table 1. Compliant Parallel Mechanism

Configurations with 1 to 6 dof

Translation

Rotation 0 1 2 3

0

1

2

3

The required motion can be decomposed into location of the line vector and elliptical surface as in equation 7.

[ ]( )

+

+°°°−=0

0

001

015 ,10 ,51 aM εε

[ ]( )

+

−+°°°−=00

sin

cossin0

h ε10 ,10 ,52

θε

θθ

aM

(7)

where the h is the translational pitch variable.

The first motion, M1, represents the change in orientation of the line L as shown in figure 8. Therefore, the direction vectors in motion (M2) are ( )Tθθ cos,sin,0 − , where θ is the current magnitude of M1.

Figure 8. Workspace requirements

From the definition of the dual vector, the dual part of the vector changes accordingly. Since M2 in equation (7) has two dof and the pitch is neither a pure rotation nor a pure translational motion, it should be decomposed further as,

[ ]( )

+

+°°°−=0

0

001

015 ,10 ,51 aM εε

[ ]( )

+

−+−=00

sin

cossin0

10mm0 5mm,2 mm,252

θε

θθε

aM

[ ]( )

+

−+°°°−=00

sin

cossin0

0 ε10 ,10 ,52

θε

θθ

aM

(8)

Equation 8 shows the decomposed result of the required

motions which has two rotational motions (M1 and M3) and one translational motion (M2).

The first motion’s (M1) pitch is (1+ε0), which implies that

the matching joint is CR for pure rotational motion. The direction vector; (1,0,0)T, of the M1 indicates that the rotation axis is aligned to X axis. The dual part of the line vector; (0, a, 0)T, determines the location of the origin of the axis of rotation to (0, 0, 0)T.

The pitch of (M2), (0+ε1), matches the CT joint since it is pure translational motion. The direction vector of the motion, (0, -sinθ, cosθ)T, indicates that the motion axis is aligned with the Z-axis of the preceding CR joint. The location of the joint is

X

Y

Z

End Effector Locations

a

5°

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a free choice since the translational motion has a free vector. Finally, the third motion is pure rotation whose motion axis is aligned to the Z-axis of the first CR joint and the axis is orientated in the same direction as the preceding CT joint.

Thus, the constraining leg is configured as in figure 9.

Figure 9. Constraining Leg Design

As discussed in the previous section, adding the active legs

finalize the type synthesis of the parallel machine. Each joint must be optimized for desired range of motion.

DETAILED DESIGN AND KINEMATIC ANALYSIS The orientation of the platform and the active legs are determined so the active DOFs’ of the legs are independent of each other. To determine the dimension of the compliant joints, the required motion range of the joints should be calculated through forward kinematic analysis as follows.

Figure 6 shows a configuration of a CPKM with

constraining leg and one active leg. The trace of one point on the upper plate of the platform defines the desired workspace. Active leg is connected to the platform and the base at points A and B with CU joints. The range of the motion of each of the joints in the active legs can be established by analyzing the given workspace. All active legs should be designed independently through similar kinematic analysis.

The two coordinate systems O and O’ are attached to the platform and the base respectively. Since the base is fixed to the ground, coordinate system O does not move. The coordinate system O’ moves with the platform. The connecting points A and B define the position vector representing the end positions of the active leg and are also the centers of the CU joints.

The length of the active leg is defined as;

NNl BA −= . (8)

where, N is the index of the positions within the prescribed

workspace. The change in actuator length is the required input. The

displacements of the each of the CU joints in the active legs are given by:

00

0

'' OABOABOABOAB

−∠∠=∠−∠=

NNB

NA

θθ

(9)

From this inverse calculation, the range of CU joints are

established. The motion range of the CU joint in the constraining leg is calculated as;

0'' OROORO −∠∠= Nθ . (10)

From the computed motion ranges, the dimensions of the

compliant joints are established. The final design is shown in figure 10.

Figure 10. Final Design of CPKM

CONCLUSIONS The paper introduced compliant parallel kinematic

machines (CPKM) using large displacement compliant joints and presented a systematic methodology for synthesizing CPKMs.. The complaint joints used in CPKMs offer many benefits including high accuracy, high off-axis stiffness, distributed compliance avoiding high stress concentrations, and minimum drift in motion axis.

We presented a simple method of configuring a PKM of any number of degrees of freedom by first synthesizing the constraining leg and incorporating standard active legs. The

a

5°

25mm

z-axis

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configuration design (type synthesis) process reduces to selecting an appropriate kinematic chain from the library of constraining legs presented in this paper.

We presented a systematic methodology using dual vector

algebra to decompose desired arbitrary motions in 3d space in to a sequence of rotations and translations. The decomposed motions are then mapped to corresponding complaint joints of the constraining leg. A simple design example is presented to illustrate this method. The design methodology presented in this paper provides a systematic procedure to design a customized PKM that meets the desired motion specifications using standard building blocks (compliant joints, constraining legs and active legs). Using this methodology, a high precision 3-dof CPKM, to serve as a reconfigurable work support for a reconfigurable machine tool, has been designed and is now being fabricated.

Forward kinematics of PKMs has been a difficult step in

the design of conventional PKMs. Complaint PKMs simplify forward kinematics. By simply measuring the displacement of each of the compliant joints of the constraining leg with embedded sensors (not discussed in this paper), we can accurately predict the 3D position and orientation of the end-effector mounted on the platform. With closed loop control, the measured displacement (strains) can be used to control the position of the actuator. This give rise to another major benefit of CPKMs in that very high positional accuracy can be readily obtained with low precision actuators.

ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of

the NSF Engineering Research Center for Reconfigurable Machining Systems (NSF Grant EEC95-92125), University of Michigan and the valuable input from the Center's industrial partners.

REFERENCES Dasgupta, B., and Mruthyunjaya, T.S., 2000, “The Stewart

Platform Manipulator: a review”, Mechanisms and Machine Theory, Vol. 35, pp. 15-40

Hara, A., and Sugimoto, 1989, “Shyness of Parallel Micromanipulators”, ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 111, pp. 34-39

Moon, Y.-M., 2000, “Reconfigurable Machine Tool Design: Theory and Application”, Ph. D. Dissertation, University of Michigan, Ann Arbor, MI

Moon, Y.-M, Trease, B., and Kota, S., 2002, “Large Displacement Compliant Joint Design”, Submitted for Review, ASME DETC 2002

Penette, E., Henein, S., Magnani, I., and Clavel, R., “Design of Parallel Robots in Microbotics”, Robotica, Vol. 15, pp.417-420

Stewart, D., 1965, “A Platform with Six Degrees of Freedom”, Proceedings of the 1965-66 of the Institute of Mechanical Engineers part 1 180 (15), pp. 371-386