A unified approach to the accuracy analysis of planar parallel manipulators both with input uncertainties and joint clearance

Mechanism and Machine Theory 64 (2013) 1–17

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A unified approach to the accuracy analysis of planar parallel manipulators both with input uncertainties and joint clearance Genliang Chen b,⁎, Hao Wang a, b, Zhongqin Lin a, b a b

State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, 200240, PR China Shanghai Key Laboratory of Digital Manufacture for Thin-walled Structures, Shanghai Jiao Tong University, Shanghai, 200240, PR China

a r t i c l e

i n f o

Article history: Received 29 January 2012 Received in revised form 3 January 2013 Accepted 4 January 2013 Available online xxxx Keywords: Accuracy analysis Planar parallel manipulators Generalized kinematic mapping Input uncertainty Joint clearance

a b s t r a c t This paper presents a unified approach to predict the accuracy performance of the general planar parallel manipulators (PPMs) both due to the input uncertainties and the joint clearance. Based on the theory of envelope, a geometric method is employed to uniformly construct the indeterminate influences of these two error sources on the pose (position and orientation) deviation of the manipulators. According to the generalized kinematic mapping of constrained plane motions, the end-effector's exact output error bound for a specified configuration can then be obtained as an accurate and complete description for the manipulator's accuracy performance, from which not only the maximal position and orientation errors but also their coupling relationship can be derived. The planar 3-RPR manipulator is intensively studied as a numerical example and several numerical simulations are provided to demonstrate the correctness and effectiveness the proposed approach. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Parallel manipulators are widely claimed intrinsically more accurate than their serial counterparts in the literature. However, Briot and Bonev [1] have pointed out that there is no simple answer to this question of superiority. Moreover, due to the existence of the assembling and manufacturing tolerances, the active joints' input uncertainties (also known as the input errors of the actuators) and the clearance between the pairing elements of kinematic joints, unexpected displacements of the manipulator's end-effector will be inevitable in the practical applications of the parallel manipulators [2]. Therefore, it is necessary and important to develop an effective and accurate approach to predict the influences of these error sources on the position and orientation deviation of parallel manipulators. Among the above error sources, the manufacturing and assembling tolerances are considered predictable for their deterministic influences on the pose deviation and can be decreased using calibration [3,4]. Therefore, they are not taken into account in the accuracy analysis of planar parallel manipulators in this paper. On the contrary, the latter two factors lead to indeterminate effects on the manipulator's accuracy performance due to their non-predictable and non-repeatable nature, which cannot be compensated with any kind of calibration. Therefore, they become the main source of positioning errors in manipulators [5,6]. Although both of these two error sources generate unexpected motion to the parallel manipulators, their influences on the accuracy performance are exhibited in completely different ways. The input errors cause uncertainties to the kinematic inputs, which can be considered as small input ranges of the actuated joints. Consequently, in kinematics, there is no difference for this issue from the forward position analysis of parallel manipulators. Thus, it can be modeled as a problem of workspace generation for the associated manipulators and solved in many conventional methods [7,8]. On the other hand, the joint clearance introduces extra DOFs displacements between the pairing elements of the kinematic joints against their nominal motions. So, it is more complicated to exactly predict the influences of the joint clearance on the accuracy performance. For this reason, in this paper we focus on developing a unified approach to predict the effects of the input uncertainties and the joint clearance on the output errors of the general PPMs. ⁎ Corresponding author. Tel.: +86 21 3420 6786; fax: +86 21 3420 4542. E-mail addresses: [email protected], [email protected] (G. Chen). 0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.01.005

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During the past two decades, much research has been devoted to the study of accuracy analysis of parallel manipulators with input errors. As indicated by Merlet [3], the maximal position and orientation errors are considered as the best accuracy measurements for parallel manipulators. Many methods have been proposed to predict the maximal output errors. Merlet et al. [9] proposed a general method to find the largest maximal positioning error of parallel manipulators based on the classical interval analysis. Briot et al. [10] generalized the interval analysis method for the accuracy analysis for several PPMs and gave a detailed mathematical proof. The interval analysis method formulated the error model based on the forward kinematics of parallel manipulators, which is relatively computer-intensive and encounters the multiple solutions problem as well [11]. Bonev et al. [12] presented a geometric method for deriving the exact maximal pose errors for several specific PPMs. Wang et al. [13] presented an approach for the prediction of the output error bound of parallel manipulators based on the level set method. However, when the manipulator evolves both translational and rotational DOFs, it is difficult to establish an effective description for the output error vector with physical interpretations. Therefore, we [14] recently proposed a new approach for the accuracy analysis of general PPMs based on the generalized kinematic mapping of constraint plane motions. The exact output error solid is generated as an accurate and complete description for the accuracy performance. On the other hand, for the purpose of predicting the influences of the joint clearance on the output pose errors, several methods have been proposed. Lin et al. [15] used the homogeneous error transformation matrices to derive the pose errors subjected to joint clearance. Ting et al. [16] investigated the effects of joint clearance on the position and orientation deviation of planar linkages and manipulators in a geometrical method based on the rotatability laws. Uphoff et al. [4] generalized the determination of the unconstrained motion of the end-effector of parallel manipulators due to joint clearance. Parenti-Castelli et al. [17,18] proposed a novel technique for evaluating the clearance influence on the accuracy performance of parallel manipulators. Li et al. [19] formulated the accuracy analysis of parallel manipulators as a standard convex optimization problem which can predict the maximal pose errors. Recently, Binauld et al. [20] investigated on the kinematic sensitivity of robotic manipulators to the joint clearance and utilized the non-convex quadratically constrained quadratic programs to calculate the maximal pose errors of the end-effector. Among the existing literatures, most of the methods [15,17,19,20] established the error model based on the derivatives of the algebraic loop closure equations which just exhibits the first order linear approximation. Although the rotatability laws [16] successfully transformed the problem to the position analysis of planar linkages, it will become really complicated and difficult in the applications of multiple closed-loop structures. Moreover, all the existing research investigated the accuracy performance due to one aspect. None of them simultaneously investigated the uncertainty effects of these two error sources in a unified manner. In this paper, we proposed a unified approach to predict the exact output error bound of the general PPMs due to the actuators' input uncertainties and the joints' clearance. Based on the generalized kinematic mapping, the kinematics of the manipulators is mapped into the three-dimensional projective space. Thus, both the uncertain effects of the input errors on the actuated joints' nominal inputs and the joint clearance on the extra displacements against the ideal constraints can be modeled as the envelope of some plane geometries sweeping in an identified manner. Then, the exact error bound can be obtained conveniently as an accurate and complete description for the accuracy performance of the manipulators, from which not only the maximal position and orientation errors along various directions but also their coupling relationship can be derived. The model for the accuracy analysis is established in an analytical and straightforward way and the exact output error bound can be obtained conveniently and efficiently in a closed-form manner. In the end, the planar 3-RPR manipulator (here R and P denote the revolute and prismatic joints respectively and the underlined R’ means the revolute joint is actuated) is studied as a numerical example to demonstrate the effectiveness and efficiency of the proposed approach. The rest of this paper is organized as follows. In Section 2, we present a brief introduction to the generalized kinematic mapping of constrained plane motions and its application to the accuracy analysis of PPMs due to the input errors. Then, the effects of the joint clearance on the motion constraints are studied in Section 3 where their influences on the kinematic constraints of the individual legs are demonstrated by means of a simple example. Thereafter, the procedure for the generation of the exact output error bound both with input uncertainties and joint clearance are presented in Section 4, where followed by a numerical example in Section 5. In the end, a short conclusion is drawn in Section 6. 2. Generalized kinematic mapping and its application to the accuracy analysis of PPMs This paper is mainly based on the concept of generalized kinematic mapping of constrained plane motions in our previous work [14]. In order to simplify the understanding to the proposed method, a brief introduction to this concept will be given in this section, as well as the application to the accuracy analysis of the general three-legged PPMs due to the input errors of actuators. 2.1. Mathematical fundamentals of the generalized kinematic mapping As illustrated in Fig. 1, the plane displacement D of a moving body E can be determined by three independent quantities (a,b,ϕ) referred to the body-fixed frame oxy. Let OXY which coincides with the original position of oxy be the inertial frame attached to the fixed plane Σ. Then, the displacement D(a,b,ϕ) of E can be given by the homogeneous linear transformation as 32 ′ 3 3 2 ′ cosϕ − sinϕ a x X ′ 4 Y 5 ¼ 4 sinϕ cosϕ b 54 y′ 5 ′ ′ 0 0 1 Z z 2

ð1Þ

G. Chen et al. / Mechanism and Machine Theory 64 (2013) 1–17

3

Fig. 1. General displacement of constrained plane motions.

where a, b are translational components of D, which are also the coordinates of the origin of oxy expressed in OXY. While ϕ is the rotational component, namely the relative orientation of oxy with respect to OXY. (x′, y′, z′) T and (X′, Y′, Z′) T denote the homogeneous coordinates of an arbitrary point Q on E expressed in the two frames, respectively. As shown in the figure, there exists an unique point P about which the plane displacement D can be considered as a pure rotation from its “zero-position” with the rotation angle ϕ. Based on this idea, the classical kinematic mapping by Grünwald [21] and Blaschke [22] was defined to map the plane displacements onto the points in a three-dimensional projective space Σ′ as     1 1 1 1 1 1 X 1 : X 2 : X 3 : X 4 ¼ xp u : yp u : u : 1 ¼ a sin ϕ−b cos ϕ : a cos ϕ þ b sin ϕ : 2 sin ϕ : 2 cos ϕ ð2Þ 2

2

2

2

2

2

where X1, X2, X3, X4 denote the homogeneous coordinates of the displacement in the image space. u ¼ tanð12ϕÞ, xp and yp are the coordinates of the rotation center P which satisfy xp = Xp, yp = Yp and can be uniquely determined. Then, an inverse mapping can be obtained conveniently, according to which a unique pre-image (a displacement of plane motion) can be determined for any given point in Σ′ as 1 X 2ðX 1 X 3 þ X 2 X 4 Þ 2ðX 2 X 3 −X 1 X 4 Þ tan ϕ ¼ 3 ; a ¼ ;b ¼ : 2 X4 X 23 þ X 24 X 23 þ X 24

ð3Þ

Consequently, substituting the above equation into Eq. (1), the expression of plane displacements can be rewritten in terms of the homogeneous coordinates Xi(i = 1, 2, 3, 4) of its image point as 2 2 3 2 ′ X −X 3 X 1 6 4 4 Y′ 5 ¼ 4 2X 3 X 4 X 23 þ X 24 ′ Z 0 2

−2X 3 X 4 2 2 X 4 −X 3 0

32 3 ′ 2ðX 1 X 3 þ X 2 X 4 Þ x 74 ′ 5 2ðX 2 X 3 −X 1 X 4 Þ 5 y : ′ 2 2 z X3 þ X4

ð4Þ

On the other hand, the inverse transformation of Eq. (4) can be obtained readily as the displacement of the fixed plane with respect to the moving one from the viewpoint of relative motions. 2 2 2 ′3 X 4 −X 3 x 1 6 ′ 4y 5 ¼ 4 −2X 3 X 4 X 23 þ X 24 ′ z 0 2

2X 3 X 4 2 2 X 4 −X 3 0

32 3 ′ 2ðX 1 X 3 −X 2 X 4 Þ X 7 ′ 2ðX 2 X 3 þ X 1 X 4 Þ 54 Y 5: ′ 2 2 Z X3 þ X4

ð5Þ

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Eq. (2) illustrates the classical kinematic mapping of plane motions according to which the plane displacements can be uniquely transformed onto the points in a three-dimensional projective space. A very systematic and detailed account on this subject can be found in Bottema and Roth's well-known treatise [23] and many other researchers' literatures [24–26]. To derive the kinematic constraints of typical legs in PPMs, the classical mapping of plane kinematics is generalized to the case of constrained plane motions. Suppose that the moving body E be constrained by an arbitrary curve Γ at Q, as shown in Fig. 1, which means that the moving body can rotate about Q when it moves on the constraint curve. Then, the plane motion constrained by Γ can be represented by a set of constraint equations in Σ′ as 8    1 ′ 2 2 ′ ′ ′ > > X 4 −X 3 F x þ 2X 3 X 4 F y þ 2ðX 1 X 3 −X 2 X 4 Þ F z >x ¼ 2 > 2 > X3 þ X4 <     1 ′ ′ 2 2 ′ ′ y ¼ 2 −2X 3 X 4 F x þ X 4 −X 3 F y þ 2ðX 2 X 3 þ X 1 X 4 Þ F z > 2 > > X3 þ X4 > > : ′ ′ z ¼Fz

ð6Þ

where F ′ x, F ′ y and F ′ z are the homogeneous coordinates of the curve's parametric equation Γ : F ′ ðλÞ, which can be transformed to its Cartesian form as " #   1 F′x X : F ðλÞ ¼ ¼ ′ ′ Y Fz Fy

ð7Þ

By multiplying the left and right sides of Eq. (6) with coordinates as    x 2 2 ¼ X3 þ X4 y

"

2

2

X 4 −X 3 −2X 3 X 4

#  X3 2X 3 X 4 2 2 F ðλÞ þ 2 X 4 X 4 −X 3

X 23 þX 24 z′

−X 4 X3

and



X1 X2

1

F ′z

respectively, it can be transformed to the Euclidean

 ð8Þ

where (x,y) T is the corresponding Euclidean coordinate of the point of interest Q in the moving frame oxy which is a constant vector when the body moves. In a sense, the above equation can be considered as a set of two linear equations of (X1,X2) when the other two (X3,X4) are specified. Then, it can be solved easily and normalized conveniently by setting X4 = 1, which yields 

X1 X2

 ¼

 1 X3 2 1

  1 X3 −1 F ðλÞ þ X3 2 −1

1 X3

  x : y

ð9Þ

Furthermore, Eq. (9) can be rewritten in a more concise and meaningful form as 

X1 X2

 ¼

 h i  cscφ h i x R φ F ðλÞ− R ϕ y 2 X 3 ¼ cotφ

ð10Þ

    cosφ − sinφ cosϕ − sinϕ and Rϕ ¼ are defined as planar rotation matrices relating to the angle φ and ϕ, sinφ cosφ sinϕ cosϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi respectively, which can be derived as cosφ ¼ X 3 = 1 þ X 23 ; sinφ ¼ 1= 1 þ X 23 , namely φ = (π − ϕ)/2 readily. Eq. (10) represents a spatial surface in the three-dimensional projective space in terms of two independent variables φ and λ, as shown in Fig. 2(a). It is easy to find that when φ, as well as ϕ, is specified, the equation degenerates to a curve on the horizontal plane X3 = cot φ. Moreover, the generated plane curve is similar to the constraint curve Γ, which can be obtained via an affine mapping consisting of a translation (−[Rϕ](x,y) T), a rotation ([Rφ]) and a scaling ð12 cscφÞ, consequently. On the other hand, when the variable λ is fixed, the image curve of the plane motion can be specified in the following form.  where R φ ¼



X1 X2





1 ¼ 2

"

" # # λ 1 y−F λy Fx þ x X3 þ λ 2 F λx −x Fy þ y X3 ¼ X3

ð11Þ

where Fxλ, Fyλ are the coordinates of Q specified by λ. Obviously, it represents a spatial line with direction of (Fxλ + x, Fyλ + y, 2) T and intersected with the zero horizontal plane at the point (−(Fyλ − y)/2, (Fxλ − x)/2, 0) T. Moreover, it can be proved that the generated image surfaces of the constrained plane motions are ruled surfaces in the projective space. And the φ − curves are the directrix, also named as the base curve of the ruled surface and the λ − lines are the corresponding rulings or the generators of the surface.

G. Chen et al. / Mechanism and Machine Theory 64 (2013) 1–17

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Fig. 2. Image surfaces of the constrained plane motions in the projective space.

Another advantage of the parametric representation for the image surfaces is that it can deal with the plane motions constrained by non-closed curves, such as the one constrained only by half of the ellipse, as shown in Fig. 2(b). This situation is very common in mechanisms and robot manipulators due to the limitations of the actuators' capability and the passive joints' movability. It is quite convenient to cope with this situation in a parametric way proposed in our model just by restricting the curve parameter λ within a specified region, such as λ ∈ [0,π] for the first half of the constraint ellipse. Up to now, the generalized kinematic mapping for the constrained plane motions has been set up, based on which an image surface can be obtained in the three-dimensional projective space for the plane motion constrained by an arbitrary curve. As a matter of fact, this model is also applicable to the cases of area constraints. Then, the image of the constrained motion in the projective space will be a spatial solid encircled by the surface associated with the boundary of the constraint area. The detailed discussion on this approach can be found in Ref. [14].

2.2. Accuracy analysis of PPMs with input uncertainties The generalized kinematic mapping of constrained plane motions is successfully applied to the accuracy analysis of PPMs due to the input errors of the actuators [14]. In this subsection, the main procedures of this method will be briefly introduced by taking the 3-RPR manipulator (as shown in Fig. 3(a)) for an instance. The first step of the proposed method is to generate the kinematic constraints of the individual legs on the moving platform, namely the R-P-R leg in this case (as illustrated in Fig. 3(b)). Apparently, the nominal constraint for each leg on the moving

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Fig. 3. Kinematic modeling for the RPR leg with input uncertainties.

platform is a line along the axis of the prismatic joint specified by the actuated revolute joint. Then, by taking the input uncertainties into account, the actual constraint of the moving platform can be formulated as

  X ¼ ðr0 þ Δr Þ cos δ þ ε  Y ¼ ðr 0 þ Δr Þ sin δ þ ε

ð12Þ

where δ ∗ and r0 are the nominal input of the active revolute joint and the nominal length of the passive prismatic joint (namely the distance between the joints Ai and Bi), respectively. ε ∈ [− δmax, δmax] denotes the input error of the active joint and Δr is the deviation of the prismatic joint's actual length from its nominal one. Eq. (12) exerts an area constraint in the shape of sector ring (as shown in Fig. 4(a)) on the moving platform at the passive revolute joint Bi. Clearly, the two boundary radii of the sector are specified by the upper (δ ∗ + δmax) and lower (δ ∗ − δmax) limits of the input errors of the actuated revolute joint and the radii of the inner and outer arcs are corresponding to the range of the passive prismatic joint's length, namely rmin = r0 − Δr and rmax = r0 + Δr. Then, according to Eq. (10), the moving platform's kinematic constraint due to the input uncertainties of the leg can be obtained conveniently in the projective space. The resultant spatial solid is illustrated in Fig. 4(b). It is clear that the horizontal sections of the solid at different heights are all areas in the

Fig. 4. Kinematic modeling for the RPR leg with input uncertainties.

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Fig. 5. Output errors of the 3-RPR manipulator due to input errors.

shape of sector ring. And all of these sections are similar to the original one illustrated in Fig. 4(a), but different in position, orientation and dimension according to the affine mapping defined in Eq. (10). According to the consistency condition of kinematics, the undesired displacements of the moving platform can be obtained by intersecting the kinematic constraints of the individual legs in the projective space, as shown in Fig. 5(a). Since all the horizontal sections of the legs' constraint solids are in the shape of sector ring and can be specified in terms of the heights of the horizontal

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G. Chen et al. / Mechanism and Machine Theory 64 (2013) 1–17

planes, namely the coordinate X3. Then, the intersection of the constraint solids can be determined plane by plane at different levels. For each plane, the intersection of the spatial solids degenerates to a planar problem to identify the common area of the sector rings. As mentioned above, the inner and outer arcs of the sectoring depend on the length of the passive prismatic joint which can change freely due to the non-actuated property. It means that the vertices of the intersection will always lie on the upper and lower boundary radii. Finally, the problem has been transformed to the one of specifying the smallest common area enclosed by three pairs of planar lines associated with the individual legs, the filled triangular region as illustrated in Fig. 5(b). Then, the error solid of the studied manipulator can be generated readily in the projective space and conveniently transformed to the exact output error bound in the output error space (as shown in Fig. 5(c)) by virtue of the inverse mapping defined in Eq. (3). From the figure, it is obvious that the output error vector (combining the translational and rotational components) is confined within the obtained error bound, from which the maximal output error in arbitrary directions can be extracted directly. It should be noted that the sections of the constraint solids of the legs will be variant according to the exact structures of the studied manipulators. But all of them will be enclosed by simple planar curves, such as line and arcs, due to the particular structures of the planar revolute and prismatic joints. Thus, it will not be difficult to determine the intersection of the error solids at different heights of horizontal planes in the projective space, so is the whole error solid. Therefore, it can be stated that the strategy of the proposed approach is generic and available for the accuracy analysis of all potential candidates of PPMs. And the procedures for the derivation of the exact output error bound are independent from the concrete architectures of the manipulators. 3. Accuracy analysis of planar mechanisms with joint clearance From this section on, we will extend the above approach to the accuracy analysis of planar manipulators with joint clearance and then establish a unified procedure to predict the exact output error bound of the general PPMs both with input uncertainties and joint clearance. As addressed in the above section, the main idea of the presented approach is to obtain the intersection of the individual legs' motion constraints, which is regarded as the output error solid in the projective space. It has nothing to do with the causes of the motion constraints of the legs exerted on the moving platform. Therefore, it can be naturally applied to the modeling for the output errors due to joint clearance. What needs to do is just to clarify how the joint clearance influences the motion constraints of the legs. In the rest of this section, a detailed discussion on this problem will be presented based on the theory of envelope [27–29]. And then, the slider-crank mechanism, as shown in Fig. 6, will be studied as an example to demonstrate the accuracy analysis due to the joint clearance. 3.1. Kinematic modeling of the lower pairs affected by joint clearance There are two typical 1-DOF lower pairs, namely the revolute joint and the prismatic joint, in the planar mechanisms. Different from the ideal ones, there are two extra tiny unexpected motions along their constraint directions in the clearance-affected planar joints. Consequently, an influence of the joint clearance will also be exerted on the motion constraints of the links connected to the joints. 3.1.1. Clearance-affected revolute joint As indicated in the literatures [16], the clearance in planar revolute joints can be generally idealized as the differences of the radii of the pin and the hole of the pair, as shown in Fig. 7(a). Thus, body k + 1 can rotate with respect to body k about any point within the constraint circle with radius δR = rhole − rpin, rather than only the nominal pivot of the ideal joint. Then, the relative position of any point on body k + 1, such as Q illustrated in the figure, can be obtained as

X Q ¼ r δ cosðηÞ þ lQ cosðα Þ Y Q ¼ r δ sinðηÞ þ lQ sinðα Þ

ð13Þ

Fig. 6. The planar slider-crank mechanism with joint clearance.

G. Chen et al. / Mechanism and Machine Theory 64 (2013) 1–17

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Fig. 7. Kinematic modeling for the revolute joint with clearance.

where rδ ∈ [0,δR] denotes the distance of the pin's center from the one of the hole, while η ∈ [0, 2π) is the corresponding orientation of the centers. lQ represents the length of the point Q from the pin's center and α is defined as the rotation angle of the revolute joint whose region can be specified according to the movability of the joint. If the joint is active and locked by an actuator, α will be a constant. When the joint is passive, then the joint can perform a whole turn revolution, namely α ∈ [0, 2π). According to Eq. (13), the constraint area of the point of interest can be regarded as the envelope of a moving plate (with radius of δR) along an arc (centered at the center of the hole and with radius of lQ), as shown in Fig. 7(b). Based on the theory of envelopes [28], it can be easily proved that the resultant envelope is a part of the annular area when the rotation angle varies between 0 and 2π. This representation for the joint clearance applies to both the passive and the actuated revolute joints in mechanisms. The difference between them is just the types of the nominal constraint for the joints. When the joint is passive, the primary circular constraint will sweep along the whole nominal constraint and generate a complete annulus as the actual motion constraint. Otherwise, the actual motion constraint will degenerate into a circular area due to the specified rotation angle in the accurately actuated case or just a part of annulus caused by the input errors of the actuators. Finally, the influence of the clearance on the motion constraint for the revolute joint can be determined conveniently according to its exact geometry dimensions. Therefore, it is very convenient to deal with all kinds of situations for the clearance-effected revolute joints in the geometric presentation. 3.1.2. Clearance-affected prismatic joint Analogously, the kinematics for the clearance-affected prismatic joints can also be modeled in a similar way as presented in the above subsection. As illustrated in Fig. 8(a), the clearance in plane prismatic joint can be idealized as the differences of the widths of the slider (link j + 1) and the rail (link j), as the parameter dslider and drail shown in the figure. Therefore, the slider can not only translate along the nominal axis with respect to the rail, but also perform a small translation perpendicular to the axis and a tiny rotation on the plane as long as the geometry constraint of the joint is satisfied, which can be represented as 1 1 jed þ ls sineα j≤ δP 2 2

ð14Þ

Fig. 8. Kinematic modeling for the prismatic joint with clearance.

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where ed denotes the translational error of the slider perpendicular to the nominal axis, and eα represents the rotational one. ls is the thickness of the slider and δP = wslider − wrail denotes the joint clearance. Then, the position of an arbitrary point on link j + 1, such as P, can be specified as

X P ¼ d−lP sineα Y P ¼ lP coseα þ ed

ð15Þ

where lP denotes the length of point P from the nominal center of the prismatic joint on the slider, d represents the translational distance of the slider relative to the rail along the joint axis, whose region can be specified according to the movability of the joint. According to Eqs. (14) and (15), when the slider translates along the rail, the position of the point of interest P will sweep over a belt area enclosed by two lines parallel to the nominal joint axis, as illustrated in Fig. 8(b). This belt area is symmetric about its ideal constraint line and its width equals to clearance of the joint, namely δP. Similar advantages for the kinematic modeling of clearance-affected prismatic joints can be drawn as the revolute one does. Namely, it is more realistic than the constant contact model and suitable to all situations no matter the joint is passive or actuated. 3.2. Accuracy analysis of the slider-crank mechanism In this section, the slider-crank mechanism, as shown in Fig. 6, will be studied as an example to demonstrate the accuracy analysis due to joint clearance. As illustrated in the figure, the 1-DOF mechanism consists of three moving links (the crank, the coupler and the slider consequently) and four planar joints. The coupler is connected to the other two links with revolute joints (noted as Band C respectively) and then assembled to the ground through a revolute joint (labeled as A) at the crank end and a prismatic one (labeled as D) at the slider one, respectively. When the crank rotates about the pivot of A, the slider will translate along D back and forth, and the coupler will produce a particular trajectory which is synthesized for some prescribed tasks in the classical problem of dimensional synthesis of four-bar mechanisms [30]. However, when the clearance of the joints is taken into account during the kinematic analysis, variable position and orientation deviation of the coupler occurs and the mechanism cannot perform the given tasks accurately any more. At the crank end, the potential position of B on the coupler, namely the center of the hole part, can be represented as  r B′ ¼ r A þ

     r δ1 cosη1 r cosη2 l cosϕ þ 1 þ δ2 r δ1 sinη1 r sinη l1 sinϕ δ2 2

ð16Þ

where rA represents the position vector of A on the fixed frame. r δ1 ∈½0; δ1  and r δ2 ∈½0; δ2  denote the center distances of the pins and the holes of the revolute joints A and B, respectively. While δ1 and δ2 are the corresponding joint clearance. η1, η2 ∈ [0, 2π) denote the orientations of the centers. l1 and ϕ represent the length and the angular input of the crank, respectively. Since the pin can arbitrarily move relative to the hole within the joint clearance. The variables r δ1 , rδ2 , η1 and η2 can be considered as free variables only constrained by their own ranges. Then, the second and fourth items of Eq. (16) can be combined together as a larger circular area and the above equation can be rewritten as #   " ′ l1 cosϕ r δ′ cosη ð17Þ r B′ ¼ r A þ þ ′ l1 sinϕ rδ′ sinη where rδ′ ∈½0; δ1 þ δ2  and η′ ∈ [0, 2π).

Fig. 9. Motion constraint of the coupler in the projective space.

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From the viewpoint of kinematic geometry, Eq. (17) can be regarded as a circular region moving along the arc determined by rA (center), l1 (radius) and ϕ (central angle). It is completely same as the situation of Eq. (13). For any given input ϕ ∗ of the crank, the coupler will be constrained, at the center of the revolute joint B, within a circular region centered at its nominal position rB = rA + l1(cos ϕ ∗, sin ϕ ∗) T and with radius δ′ = δ1 + δ2. Then, at the crank end the coupler's motion constraint in the 3-dimensional projective space can be derived conveniently by substituting the constraint function F(λ) in Eq. (10) with rB′ . In this situation, it has been proved [14] that the unexpected displacement of the coupler will be restricted within the spatial solid enclosed by a hyperboloid of one sheet, as shown in Fig. 9. On the other hand, the coupler is also constrained at the slider end via the prismatic joint D. In the same manner, the potential position of C on the coupler can be represented as  rC′ ¼ rD þ

        r cosη3 0 cosφ − sinφ d cosφ −l3 sineα þ þ þ δ3 r δ3 sinη3 ed sinφ cosφ d sinφ l3 coseα

ð18Þ

where rD is the position vector of D on the fixed frame. φ and d denote the orientation and the displacement of the prismatic joint, respectively. eα and ed represent the angular and perpendicular displacements of the slider with respect to the rail. l3 represents the length parameter of the slider as same as lP in Eq. (15). r δ3 ∈½0; δ3  denotes the center distance of the pin and hole of C, and δ3 is the corresponding clearance. η3 ∈ [0, 2π) denotes the orientations of the centers of joint C. Eq. (18) can be rewritten as follows by decomposing the clearance-affected error vector along the nominal axis of the prismatic joint.     rC ′ ¼ rD þ d−l3 sineα þ r δ3 cosη3 u þ l3 coseα þ ed þ rδ3 sinη3 u⊥

ð19Þ

where u = (cos φ, sin φ) T and u⊥ = (− sin φ, cos φ) T represent the unit vectors parallel and perpendicular to the prismatic joint, respectively. Since the prismatic joint D is passive, its displacement d can be arbitrarily assigned. As a consequence, the error component  −l3 sineα þ r δ3 cosη3 doesnot influence the position along the joint. Then, the constraint region of rC ′ can be generated by sweeping the line segment l3 coseα þ ed þ r δ3 sinη3 u⊥ along the line rD + d ⋅ u. According to Eq. (15), the constraint of the error variables can be modeled as

ð−δ4 −2ed Þ=ls ≤ sineα ≤ ðδ4 −2ed Þ=ls −δ4 =2 ≤ed ≤δ4 =2

ð20Þ

where ls represents the width of the prismatic joint's slider and δ4 is its clearance. Substituting Eq. (20) into Eq. (19), the range of the error component perpendicular to the joint can be obtained readily as   l3 −δ4 =2−δ3 ≤ l3 coseα þ ed þ r δ3 sinη3 ≤l3 þ δ4 =2 þ δ3 :

Fig. 10. Accuracy performance of the slider-crank mechanism due to joint clearance.

ð21Þ

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Therefore, at the center of the revolute joint C, the coupler will be constrained within a belt region parallel to the direction of D with the maximal and the minimal distances as dismax = l3 + δ4/2 + δ3 and dismin = l3 − δ4/2 − δ3, respectively. The situation of this case is same as the one shown in Fig. 8(b) except for the determination of the belt region's width. Then, the coupler's motion constraint at the slider end can be derived conveniently in the same manner as the crank end does, which is illustrated in Fig. 9. This time, the unexpected displacement of the coupler will be bounded by two pieces of hyperbolic paraboloid associated with the two boundaries of the constraint region. For an arbitrary input of the actuated revolute joint, the unexpected displacements of the coupler can be obtained by intersecting its motion constraints due to the crank end and the slider end. Since the boundaries of the error solids at the two ends is either hyperboloid of one sheet or hyperbolic paraboloid represented in an explicitly parametric way as Eq. (10), the error solid can be derived conveniently. The result is shown in Fig. 10(a), from which the maximal rotational error can be derived directly as the extreme values along the X3 axis. As well, the maximal translational errors in various directions can be obtained by transforming the error solid from the projective space to the Euclidean plane. It generates a closed curve as the exact translational error bound of the mechanism at the given configuration, as shown in Fig. 10(b), and the maximal positioning errors in various directions are confined within it. So far, the accuracy analysis of the slider-crank mechanism with joint clearance has been accomplished. We have obtained the maximal rotational and translational errors in various directions. The analysis is based on the forward kinematics of the mechanism so that the maximal output errors are exactly accurate solutions to the accuracy analysis rather than approximate ones derived with matrix method. 4. Accuracy analysis of PPMs both with input uncertainties and joint clearance In the above section, a geometric strategy has been presented for the derivation of the exact output errors of planar mechanisms due to joint clearance. Comparing with the one used for the accuracy analysis caused by input uncertainties in Section 2.2, there is no difference between these two cases except for the causes of the motion constraints. In both cases, the derivations for the motion constraints of plane kinematics are based on the sweeping of plane curves along particular trajectories. From the viewpoint of plane geometry, the generation of the motion constraints influenced by the input uncertainties and the joint clearance can be considered identical. Therefore, a unified procedure for the accuracy analysis of the general PPMs both with these two error sources will be presented in this section. For the convenience of discussion, but without loss of generality, we still take the 3-RPR PPM for an instance. This time the motion constraints of the moving platform is not only influenced by the input uncertainties of the actuated revolute joints but also affected by the clearance in all joints regardless of whether they are passive or actuated. In general, each of the RPR leg in the manipulator can be considered as a planar 3-DOF serial mechanism, as shown in Fig. 11. The moving platform is regarded as the mechanism's end-effector which connected to the fixed one through the upper and the lower links via a revolute joint, a prismatic joint and a revolute joint again consequently. All the joints are supposed to be effected by the clearance due to which undesired tiny displacements will be produced against their nominal configurations. Therefore, the pin (here it is assumed that the pin is attached to the lower link in the limb) of the actuated revolute joint is constrained within a circular area Λ0, as shown in Fig. 12. Obviously, Λ0 is centered at its nominal pivot H0 and with the radius of εr, namely the clearance of the joint. Then, the motion constraint for the center point of the socket can be obtained conveniently by transforming Λ0 from H0 to H1 along the nominal axis of the prismatic joint, as shown in the figure. By applying the effects of the actuated joint's input uncertainties, namely εa, the potential position of the socket center of the prismatic joint and the extreme configurations of its axis can be derived readily as the area Λ1 and the lines Γ1 respectively according to the strategy presented in Section 3.1. So far, the influence of the clearance and input uncertainty in the first revolute joint on the motion constraint of the lower link has been obtained. Thereafter, it naturally comes to the investigation on the influence of the clearance in the passive prismatic joint on the motion constraint of the moving platform. However, it is a little different from the case studied in Section 3.2. Rather than fixed, the position and orientation of this prismatic joint can move freely within its motion constraint as Λ1 and Γ1. Then, the pivot of the passive revolute joint's hole (on

Fig. 11. The RPR leg both with input uncertainties and joint clearance.

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13

Fig. 12. Motion constraint of the RPR leg both affected by input uncertainties and joint clearance.

the upper link) will be confined within an area enclosed by the polylines Γ2, as illustrated in the figure, according to the geometry dimensions of the passive prismatic joint. It should be noted that the line B1 C 1 intersects with D2 E2 at the point B1 because the extreme configurations of the prismatic joint are not parallel any more. Thus, the boundary of Λ2, namely Γ2, are broken lines consisting of four line segments rather than three in the case of fixed prismatic joints. Thereinto, the line segment C 1 D1 is offset from Γ1 at the distance of εp (the clearance of the prismatic joint) and has the same length with the socket, lsocket. The angle α between the line D1 E1 (or D2 E2) and Γ1 is determined in terms of the maximal rotational error of the prismatic joint. And the line A1 B1 is collinear with D2 E2 as the extreme position for the hole in the opposite direction of the prismatic joint. Simultaneously, the clearance in the last revolute joint also performs a circular constraint to the moving platform. Then, the pin on the moving platform is constrained within a circular area with radius of εr and centered within Λ2. Finally, an area, denoted as Λ3 enclosed by the broken lines Γ3, is obtained by offsetting Γ2 outward at a distance of εr as the final motion constraint for the moving platform. Λ3 is a complete description for the constraint of the moving platform under the assumption that the passive prismatic joint can translate to infinity along its nominal axis. Actually, for a given configuration it only produces a small deviation for the length of the passive prismatic joint due to the joint clearance and the input uncertainties. Therefore, a small area at the neighborhood of the nominal position can be extracted from the complete motion constraint, which is sufficient for the accuracy analysis of PPMs. Consequently, the motion constraint for the moving platform in the studied example can be updated, relative to the one obtained in Section 2.2, by taking the joint clearance into account. And for the same configuration, the motion constraint can be finally obtained and illustrated in Fig. 13 with the one only due to input errors as a comparison. From the figure, it is obvious that the constraint area has been enlarged due to the clearance of the kinematic joints. Meanwhile, the figure also intuitively demonstrates how the clearance in the three joints influences the motion constraint on the moving platform in a geometric way. On one hand, the clearance in revolute joints enlarges the constraint area by offsetting the two radii of the sector-ring outward. On the other hand, the clearance in the prismatic one increases the central angle of the sector and makes the two sector-rings for the motion constraints not concentric any more. Thus, it is obvious that the motion constraint of the moving platform is also affected by the length of the lower link, namely the assembling position of the prismatic joint in the leg. This is different from the general understanding in the case of non-clearance affected situations that the length of the leg refers to the distance between the pivots of the two revolute joints on the moving and the fixed platforms, respectively. Finally, the motion constraints corresponding to the individual legs can be obtained in the image space according to the generalized kinematic mapping. And the unexpected displacements of the moving platform can be derived as the error solid in the image space by intersecting the constraints, which can be transformed to the exact output error bound of the end-effector in the error space via an inverse mapping. By now, the accuracy analysis of the 3-RPR parallel manipulator both with joint clearance and input uncertainties has been accomplished. A complete and accurate description for the maximal output errors, combining with the translational and rotational ones, has been obtained as a spatial solid in the error space, from which all the information on the accuracy performance of the manipulator can be derived. Moreover, from the discussion it is obvious that the approach is independent of the concrete architectures of the manipulators. In other words, it can be applied to any potential candidate of the general three-legged PPMs in the same manner. 5. Numerical examples and discussions In this section, the above 3-RPR manipulator will be studied again as a numerical example for the accuracy analysis of PPMs both with input uncertainties and joint clearance. The specification of the manipulator's geometric parameters, apart from the joint

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Fig. 13. Gradual influences of the joint clearance and input uncertainties on the motion constraint of the moving platform.

clearance, is same as the one studied in Ref. [10]. The clearance of the revolute joints is given by εr = 1.0× 10−6 m. For the passive prismatic joints, the clearance between the slider and the rail is supposed to be εp = 1.0 × 10 −5 m and the thickness of the socket is assigned as ls = 0.02 m. While the input uncertainties of the actuators are confined within the bound of |εa| ≤2.0 × 10 −4 rad. The length of the lower link is settled as llower = 0.2 m and llower = 0.1 m, respectively, for the purpose of investigating on the influence of the assembling position of the prismatic joints on the accuracy performance of manipulator. The configuration of the studied manipulator can be assigned arbitrarily within its dexterous workspace. For the convenience of calculation, it is specified at the geometry center of the manipulator without any rotation, namely x = 0, y = 0, ϕ = 0. Then, the results can be obtained according to the method discussed in the above sections. The output error solid in the image space are illustrated in Fig. 14(a), where a comparison is made to the case only caused by the input uncertainties of the actuated joints as presented in Section 2.2. As well, the contours for the translational output errors at different levels of the rotational ones are drawn in the figure as an intuitive description for the accuracy performance of the manipulator. From the figure, it is obvious that the maximal output errors, both the translational and the rotational ones, of the manipulator are enlarged due to the existence of the joint clearance, although the shape of the error solids are nearly same. The magnitude of the maximal rotational errors of the end-effector increases from 0.50 × 10 −3 rad to 0.77 × 10 −3 rad. And the maximal translational ones are also extended from 0.667 × 10 −4 m to 1.03 × 10 −4 m.

Fig. 14. Accuracy performance of the 3-RPR manipulator with and without joint clearance.

G. Chen et al. / Mechanism and Machine Theory 64 (2013) 1–17

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Moreover, for each point in the image space there exists a corresponding pre-image on the motion plane as the tiny displacement for the moving platform. Then, the kinematic constraints of the individual legs on the moving platform can be determined exactly for each maximal output error point, and the contact conditions for all joints with clearance can be determined in terms of the geometric dimensions and the concrete structure of the consisting legs. Take the maximal rotational error (0.77 × 10 −3 rad) as an example. As illustrated in the figure, the intersection point for each motion constraint caused by the individual legs can be determined for this configuration. Then, it is convenient to specify the corresponding point in the motion constraint for the pin of the passive revolute joint on the moving platform, as the point A3 in Fig. 13. In a consequence, the actual position of the passive revolute joint's hole, the center of the passive prismatic joint's socket and the pin of the actuated joint can be determined as the point A2, A1 and A0 accordingly. Thus, it is obvious that the clearance error of the actuated joint is O0 A0, namely the two paring elements in this joint is in contact at the point A0. Analogously, the input error of this joint can also be determined as εa. While, the clearance error in the prismatic joint is ed = 0, eα = α, of which the deviation of the length along its nominal axis can also be specified readily as Δl ¼ A1 A2 −lup . Similarly, the clearance error in the passive revolute joint can be represented as the vector A2 A3 . Therefore, for all potential output errors, the influences of the joint clearance and the input errors on the position and orientation's deviation of the end-effector can be specified in the same manner conveniently. Meanwhile, another comparison is made for the output errors with different lengths of the lower link (namely llower = 0.2 m and llower = 0.1 m respectively). The numerical results are illustrated in Fig. 15. From the figure, it is clear that the change of the lower link's length produces a palpable effect on the output errors of the manipulator. The maximal rotational one has been changed from 0.77 × 10 −3 rad to 1.27 × 10 −3 rad, and it is from 1.03 × 10 −4 m to 1.69 × 10 −4 m for the maximal translational ones. This result indicates that the concrete structure of the links greatly influences the accuracy performance of PPMs with joint clearance. Although the architecture and geometry dimensions of the manipulator are completely same, the output accuracy will be different due to the assembling position of the passive prismatic joints. Additionally, the deviations of the passive joints' actual motion to their nominal ones are derived for the three different cases according to the aforementioned strategy. The values for the maximal output error conditions are enumerated in Table 1. In the table, we can see that these deviations are rather big (5.70 × 10 −4 rad and 4.40 × 10 −4 m) comparing with the input uncertainty of the actuators and the clearance in kinematic joints. Therefore, when the individual leg is extracted from the whole manipulator as a serial chain, the output errors of the end-effector is not only effected due to the input errors and the joint clearance but also influenced by the deviations of the passive joints' nominal configurations which seems relatively bigger than the nominal one. Furthermore, the magnitudes in the third case are much bigger than the first two, which is in consistence with the situation of the output errors. This phenomenon can explain, to some extent, why the parallel manipulators are not as accurate in practical applications as the ones analyzed theoretically. It is generally acknowledged that the kinematic errors caused by different sources will not be accumulated at the end-effecter in the parallel manipulators due to their multiple closed-loop structure, just opposite to their serial counterparts. However, the statement has been made [5] and verified [19] that the accuracy performance of parallel manipulators is highly sensitive to certain clearance in the kinematic joints. As a matter of fact, due to the existence of the massive passive joints, the error-caused kinematic conflicts between the legs would be compromised by means of changing the nominal configurations of the passive joints. Moreover, the original structure errors may even be amplified due to the passive movements of the underactuated joints, which will worsen the parallel manipulators' accuracy performance. Therefore, much attention should be paid on the elimination of the joint clearance, such as by preloading the pairing elements. In the end, as indicated in the above, using the proposed approach the accuracy analysis can be performed at arbitrary configurations within the prescribed workspace. Therefore, the accuracy performance of the studied manipulator within a circular

Fig. 15. Accuracy performance of the 3-RPR manipulator with different assembling positions of the passive prismatic joints.

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Table 1 Deviations of the nominal configurations of the kinematic joints, both the clearance errors and the passive motions. Output

Cases

Actuated revolute joint Clearance

Rotational

Clearance

Translational

Clearance

I II

– 

εα εα

– (α,0)

1.73 × 10−4 m 2.67 × 10−4 m

– 

εα

(α,0)

4.40 × 10−4 m

εα εα

– (α,0)

1.15 × 10−4 m 1.78 × 10−4 m

εα

(α,0)

2.93 × 10−4 m

Error type Maximal rotational error

III Maximal translational error

I II III

  ε γ cos ε a þ π2 π ε γ sin εa þ 2    ε γ cos ε a þ π2 ε γ sin εa þ π2 –    ε γ cos ε a þ π2 ε γ sin εa þ π2    ε γ cos ε a þ π2 π ε γ sin εa þ 2

Passive prismatic joint

Passive revolute joint

  εγ cos εa þ α þ π2 π εγ sin εa þ α þ 2    εγ cos εa þ α þ π2 εγ sin εa þ α þ π2 –    εγ cos εa þ α þ π2 εγ sin εa þ α þ π2    εγ cos εa þ α þ π2 π εγ sin εa þ α þ 2

Rotational 3.0 × 10−4 rad 0.7 × 10−4 rad 5.7 × 10−4 rad −0.34 × 10−4 rad −4.43 × 10−4 rad −2.77 × 10−4 rad

workspace whose radius is equal to 0.20 m for orientation angle ϕ = 0 is studied. In this case, the lengths of the lower link in the legs are set to 0.1 m and the specification of the geometric parameters is as same as above. The whole workspace is discretized into some grids and the accuracy analysis is performed point by point. Then, the maximal rotational errors within the prescribed workspace, as illustrated in Fig. 16(a), can be obtained from the exact output error bounds. On the other hand, the maximal output errors only due to the input uncertainties can be obtained conveniently in our model by setting the clearance of the kinematic joints to zeros, which is also illustrated in Fig. 16(a) as a comparison. The contours of the maximal rotational errors within the given workspace are drawn in Fig. 16(b) which can be validated by the results obtained in Ref. [10]. The maximal translational errors can be obtained in the same manner. 6. Conclusions In this paper, the accuracy analysis is achieved for the general PPMs subjected to the errors due to the input uncertainties of the actuators and the clearance of the kinematic joints. The obtained error bound can be considered as a tiny workspace, which means that the solution is a more accurate and complete description for the accuracy performance than the linear approximation by the Jacobian matrix. Using the proposed approach, except for the exact output error bound at a specified configuration, the detailed contact modes of all clearance-affected joints can also be obtained conveniently for the worst situations of the manipulators' accuracy performance, so is the maximal output errors within a prescribed workspace. From the numerical results, some characteristics about the accuracy performance of the general PPMs are outlined, which can be served as design criteria or explanation to the undesired accuracy performance of parallel manipulators, such as the statement that the concrete structure of the individual legs influences the maximal output errors greatly and the existence of the free motions in the passive joints is considered as one of the key factors to the undesirable accuracy performance in practical applications. Acknowledgment This research work is jointly supported by the National Science Foundation of China (NSFC) under Grant Nos. 51075259 and 51121063 and the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education of China.

Fig. 16. Maximal rotational errors for the 3-RPR manipulator within the prescribed workspace.

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References [1] S. Briot, I.A. Bonev, Are parallel robots more accurate than serial robots? CSME Transactions 31 (4) (2007) 445–456. [2] J.P. Merlet, Computing the worst case accuracy of a PKM over a workspace or a trajectory, The 5th Chemnitz Parallel Kinematics Seminar, Chemnitz, Germany, 2006, pp. 83–96. [3] J.P. Merlet, Jacobian, manipulability, condition number, and accuracy of parallel robots, Journal of Mechanical Design 128 (1) (2006) 199–206. [4] P. Voglewede, I. Uphoff, Application of workspace generation techniques to determine the unconstrained motion of parallel manipulators, Journal of Mechanical Design 126 (2) (2004) 283–290. [5] C. Han, J. Kim, J. Kim, F. Park, Kinematic sensitivity analysis of the 3-UPU parallel mechanism, Mechanism and Machine Theory 37 (8) (2002) 787–798. [6] M.J. Tsai, T.H. Lai, Kinematic sensitivity analysis of linkage with joint clearance based on transmission quality, Mechanism and Machine Theory 39 (11) (2004) 1189–1206. [7] G.R. Pennock, D.J. Kassner, The workspace of a general geometry planar three degree of freedom platform-type manipulator, Journal of Mechanical Design 115 (2) (1993) 269–276. [8] J.P. Merlet, C.M. Gosselin, N. Mouly, Workspace of planar parallel manipulators, Mechanism and Machine Theory 33 (1–2) (1998) 7–20. [9] J.P. Merlet, D. Daney, Dimensional synthesis of parallel robots with a guaranteed given accuracy over a specific work-space, Proceedings of 2005 IEEE International Conference on Robotics and Automation, Barcelona, Spain, 2005, pp. 942–947. [10] S. Briot, I.A. Bonev, Accuracy analysis of 3-DOF planar parallel robots, Mechanism and Machine Theory 43 (4) (2008) 445–458. [11] M. Raghavan, The Stewart platform of general geometry has 40 configurations, Journal of Mechanical Design 115 (2) (1993) 277–281. [12] A. Yu, I.A. Bonev, P. Murray, Geometric approach to the accuracy analysis of a class of 3-DOF planar parallel robots, Mechanism and Machine Theory 43 (3) (2008) 364–375. [13] H. Wang, G.L. Chen, Y. Zhao, Z.Q. Lin, Output error bound prediction of parallel manipulators based on the level set method, Mechanism and Machine Theory 45 (8) (2010) 1153–1170. [14] G.L. Chen, H. Wang, Z.Q. Lin, Generalized kinematic mapping of constrained plane motions and its application to the accuracy analysis of general planar parallel robots, Mechanism and Machine Theory 50 (1) (2012) 29–47. [15] P.D. Lin, J.F. Chen, Accuracy analysis of planar linkages by the matrix method, Mechanism and Machine Theory 27 (5) (1992) 507–516. [16] K.L. Ting, J.M. Zhu, D. Watkins, The effects of joint clearance on position and orientation deviation of linkages and manipulator, Mechanism and Machine Theory 35 (3) (2000) 391–401. [17] S. Venanzi, V. Parenti-Castelli, A new technique for clearance influence analysis in spatial mechanisms, Journal of Mechanical Design 127 (3) (2005) 446–455. [18] V. Parenti-Castelli, S. Venanzi, Clearance influence analysis on mechanisms, Mechanism and Machine Theory 40 (13) (2005) 1316–1329. [19] J. Meng, D.J. Zhang, Z.X. Li, Accuracy analysis of parallel manipulators with joint clearance, Journal of Mechanical Design 131 (1) (2009) 011013, (9 pages). [20] N. Binaud, P. Cardou, S. Caro, P. Wenger, The kinematic sensitivity of robotic manipulators to joint clearances, Proceedings of ASME Design Engineering Technical Conferences, August 15–18, 2010, Montreal, QC., Canada, 2010. [21] J. Grünwald, Ein abbildungsprinzip welches die ebene Geometrie und kinematik mit der räumlichen geometrie verknüpft, Sitzungsberichte Akademie Wissenschaften Wien 120 (1911) 677–741. [22] W. Blaschke, Euklidische kinematik und nichteuklidische geometrie, Zeitschrift für Angewandte Mathematik und Physik 60 (1911) 203–204. [23] O. Bottema, B. Roth, Theoretical Kinematics, Dover, New York, 1979. [24] J.M. McCarthy, An Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA, 1990. [25] J. Wu, A. Purwar, Q.J. Ge, Interactive dimensional synthesis and motion design of planar 6R closed chains via constraint manifold modification, Proceedings of the ASME 2009 International Design Engineering Technical Conferences, San Diego, CA, USA, 2009. [26] M.J. Hayes, M.L. Husty, On the kinematic constraint surfaces of general three-legged planar robot platforms, Mechanism and Machine Theory 38 (5) (2003) 379–394. [27] M. do Carmo, Differential Geometry of Curves and Surfaces, Metallurgical Industry Press, Beijing, 2005. (in Chinese). [28] J. Argyris, F.L. Litvin, Q.M. Lian, S.A. Lagutin, Determination of envelope to family of planar parametric curves and envelope singularities, Computer Methods in Applied Mechanics and Engineering 175 (1–2) (1999) 175C187. [29] D. Blackmore, M.C. Leu, L.P. Wang, The sweep-envelope differential equation algorithm and its application to NC machining verification, Computer-Aided Design 29 (9) (1997) 629–637. [30] C.H. Suh, C.W. Radcliffe, Kinematics and Mechanism Design, Wiley, New York, 1978.