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An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance properties matrix
MAMT-02762; No of Pages 17 Mechanism and Machine Theory xxx (2016) xxx–xxx
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An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance properties matrix Haitao Liu a, Andrés Kecskeméthy b,⁎, Tian Huang a,c a b c
Key Laboratory of Mechanism Theory and Equipment Design of The Ministry of Education, Tianjin University, Tianjin 300072, China Chair of Mechanics and Robotics, University of Duisburg-Essen, Duisburg 47057, Germany School of Engineering, The University of Warwick, Coventry CV4 7AL, UK
a r t i c l e
i n f o
Article history: Received 13 October 2015 Received in revised form 27 July 2016 Accepted 2 August 2016 Available online xxxx Keywords: Screw theory Reciprocal screw system Invariance properties matrix
a b s t r a c t This paper describes an automatic procedure for identifying a set of (6− f) natural constraint wrenches, termed natural reciprocal screws, for fb 6 degrees of freedom serial kinematic chains featuring special geometric properties. By a natural constraint wrench we mean a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution. This paper regards such R/P serial chains with corresponding special geometries. The approach is based on the concept of the invariance properties matrix (IPM), which was proposed in the past to find the constraint conditions for single-loop spatial mechanisms allowing for closed-form solutions. The method is further developed here to identify the invariant geometric properties associated with the natural constraint wrenches. Several typical lower-mobility serial chains are taken as examples to demonstrate the effectiveness of the approach. © 2016 Published by Elsevier Ltd.
1. Introduction Screw theory has been widely used for the kinematic and kinetostatic analyses of both serial and parallel mechanisms [1–5], and is considered to be an elegant and general mathematic tool for tackling first-order mobility and constraint problems in mechanisms. For an f (f b 6) degrees of freedom (DOF) serial kinematic chain, known as a lower mobility serial chain, the infinitesimal joint motion twists of the end-effector (“joint twists” for short) associated with its f 1-DOF joints form an f-screw system, while the constraint wrenches which are reciprocal to all possible infinitesimal joint twists of the chain form a (6− f)-screw system [6]. Since the joint twists can be obtained once the types of the joints and the configuration of the serial chain are given, the identification of the constraint wrenches becomes the main task of the concerned studies. From the concept of reciprocal screws introduced by Ball [7], if a wrench acts on a rigid body in such a way that it produces no work while the body is undergoing an infinitesimal twist, the two screws are said to be reciprocal [6]. That results in the zero ⁎ Corresponding author. E-mail address: [email protected] (A. Kecskeméthy).
http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002 0094-114X/© 2016 Published by Elsevier Ltd.
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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scalar product of any joint-twist screw $ =(S1 S2 S3 S4 S5 S6)T and the corresponding reciprocal screw $r = (Sr1 Sr2 Sr3 Sr4 Sr5 Sr6)T as $∘r ¼ 0
ð1Þ
where “∘” represents the reciprocal product of two screws. The relationship can be expanded to a set of homogeneous equations represented by the set of given linearly independent twists and its corresponding mutual unknown constraint wrenches. Hence, mathematically, the identification of constraint wrenches of a serial chain can be converted into the issue of determining the null space of a set of homogeneous equations. While the null space can be found by using different numerical methods such as Gauss Seidel elimination [8], Gram Schmidt orthogonalization [8], and an affine augmentation method [9,10], the reciprocal screws evaluated in this form are in the form of numerical values of its components, and it is not always evident how the constraint wrenches are related to geometric properties of the serial chain. Moreover, when the constraint wrenches form a subspace, an aleatory single set of independent constraint wrenches is obtained, which may not unveil clearly the intrinsic properties of the generators of the wrench subspace. An alternative approach to determine the reciprocal screws is to analyze the kinematic properties of some frequently used pairs and dyads, known as the observation method [6,11]. In particular, the reciprocal screws for all system of screws (i.e., the 1-, 2-, 3-, 4-, and 5-screw system) have been exhaustively studied by Hunt [12]. Moreover, Kim and Chung presented an analytic formulation of reciprocal screws by solving an appropriate set of algebraic equations without considering the geometry of a screw system [13]. By interpreting the reciprocal product of a wrench on a twist as a linear functional, a novel approach to determine the unknown twist/wrench subspaces of lower mobility serial kinematic chains was proposed in [14]. This allows one to automatically obtain a set of geometrically meaningful and visualizable constraint wrenches in an effective manner by taking into account three engineering constraints. Although the explicit expressions of constraint wrenches associated with a given kinematic chain can thus be achieved, these methods require a good knowledge of kinematics and screw theory, and especially they are time consuming for chains having complicated structures. Therefore, there is still a great demand for computer-aided tools for the automatic identification of the reciprocal screw system of a serial kinematic chain. For a serial lower-mobility kinematic chain, there exist in general geometric properties between adjacent or non-adjacent joints that are invariant with respect to the finite motion of the chain and/or its joints, such as intersection points, parallel and/or perpendicular conditions, and line-line distances. of several joint axes. When these invariant geometric properties are elementary, they can be defined as certain geometric measures between characteristic pairs of joints, as described in Hiller and Woernle [15], and then used to develop an effective approach for solving the inverse kinematic problem of robots. Further, by introducing the invariance properties matrix (IPM) to identify the characteristic pair of joints, Kecskeméthy and Hiller proposed a fully-automatic procedure for generating closed-form position kinematics solutions in the case of recursively solvable singleloop chains [16], where the measures between the characteristic pair of joints are interpreted as invariant characteristic projections of relative transformation matrices within branches of the loop onto scalar numbers. From the viewpoint of screw theory, the characteristic pair of joints actually indicates that constraint wrenches are invariant to motions within a serial kinematic chain. Drawing on this, this paper develops a novel and simple procedure to identify a natural reciprocal screw system, termed also a natural set of constraint wrenches, of serial lower-mobility kinematic R/P chains featuring special geometric conditions. By a natural constraint wrench we mean a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution, such as ad-hoc recognition of SE(3) submanifolds of motion of the end-effector by taking into consideration that the constraint wrenches are normal to the constraint hypersurfaces, as well as the improvement of efficiency in inverse or direct dynamics. The approach is related to the problem of finding persistent screw systems of a chain. While in the general case the task of finding persistent screw systems requires more involved algorithms using analytical expressions for each type of screw system, as developed by Gibson, Hunt, Rico, Duffy and Carricato [17–19], the proposed approach aims at providing a simple and easily applicable algorithm that can be used in cases of typical lower mobility chains with special architectures comprising only standard prismatic and/or revolute joints, such as they occur in practical applications. The rest of the paper is organized as follows: following a brief summary of the geometric properties of the projection operators involved, the formulation of IPM is described. Subsequently, the algorithm for the automatic detection of constraint wrenches of a twist system is presented. Finally, the application of the proposed approach is illustrated by typical serial kinematic chains frequently used in parallel mechanisms. While in this paper the basic procedure is illustrated on a selected set of examples, the algorithm can be generalized to cover more general cases, which may be pursued in further work. 2. Invariance properties matrix In order to define the invariance properties matrix (IPM), the algebraic and notational properties of homogeneous transformations are recollected first. Following the established convention in [20], the relative pose, i.e. rotation and translation, between two reference frames is represented by homogeneous transformation matrices which serve as transformation operators for points as well as free vectors. The general transformation law of an element “ξ” of the affine space (point or direction) is 0
ξ ¼ Aξ; A ¼
R r 0 1
ð2Þ
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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where R is the rotation matrix (actively) mapping the space at the base of the transformation to the space at its tip and transforming coordinates with respect to the tip frame to coordinates with respect to the base frame, r is the (active) translation of the origin of the base frame to the origin of the tip frame, in coordinates of the tip frame, and Rot[A] ≡R and Trans[A] ≡ r are extraction operators which can be used to select only the rotation or the translation part of the homogeneous transformation matrix A. Note that the three unit orientation vectors ei (i = 1 , 2, 3) in the direction of the coordinate axes and the position vector o representing the origin of the base coordinate system span a canonical basis of ℝ4: 0 1 1 B0C B ; e1 ¼ @ C 0A 0
0 1 0 B1C B e2 ¼ @ C ; 0A 0
0 1 0 B0C B e3 ¼ @ C ; 1A 0
0 1 0 B0C B o¼@ C 0A 1
ð3Þ
Then, the six elementary transformations can be given as follows 2
1 0 0 6 0 cθ −sθ 6 Rot½e1 ; θ ¼ 4 0 sθ cθ 0 20 0 cθ 0 sθ 6 0 1 0 Rot½e2 ; θ ¼ 6 4 −sθ 0 cθ 0 0 2 0 cθ −sθ 0 6 sθ cθ 0 Rot½e3 ; θ ¼ 6 40 0 1 0 0 0
3 0 07 7; 05 13 0 07 7; 05 13 0 07 7; 05 1
2
1 60 6 Trans½e1 ; s ¼ 4 0 20 1 60 Trans½e2 ; s ¼ 6 40 20 1 60 6 Trans½e3 ; s ¼ 4 0 0
0 1 0 0 0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0 0 0 1 0
3 s 07 7 05 13 0 s7 7 05 13 0 07 7 s5 1
where sθ (cθ) represents sinθ (cosθ). Hereafter, the elementary transformation will be denoted by AE(ei; σ; δ), translational and rotational transformations being distinguished by a Boolean variable σ (for the case of translation, its value is “1”; for the case of rotation, its value is “0”), while δ is the translational distance or rotational angle corresponding to the type of transformation. It is well-known that these elementary transformations form a basis for the group of rigid-body motions, meaning that any rigidbody motion can be theoretically decomposed into a sequence of these six transformations. By investigating the elementary transformations, it can be found that some geometric properties of the frame KE associated with AE remain unchanged after the transformation. For example, AE(e1; 0; θ) leaves the origin O of KE , the rotational axis coincident with e1, as well as the plane passing through point O and normal to e1 invariant, while AE(e2; 1; s) leaves any line along e2 as well as the two planes passing through point O and normal to e1 and e3 invariant (see Fig. 1). Hereby, it is known that the set of objects that are kept invariant by a transformation forms a group, termed the isotropy group [16]. Drawing on this property, the invariant properties related to a sequence of elementary transformations can be written in a matrix, termed the invariance properties matrix (IPM), in which the columns correspond to the individual transformations, and the rows correspond to the geometric elements which may have invariance properties. An entry is denoted by “1” if the corresponding geometric element of the row is invariant with respect to the transformation associated to the column, and otherwise by “0”. For illustration purposes, a non-rigorous example of an IPM is given below. Given a sequence of transformations A ¼ AE1 ðe1 ; 0; θ1 ÞAE2 ðe3 ; 1; s2 ÞAE3 ðe1 ; 0; θ3 ÞAE4 ðe2 ; 0; θ4 Þ
e3 Rotation
ð4Þ
1
e3
Translation
s
θ O
1
e2
O
e1
e2
e1 3 Fig. 1. Invariant properties of elementary transformations.
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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the corresponding IPM can be formulated as
ð5Þ
where the zeros have been omitted for better clarity. Here, O is the origin of the corresponding reference frame KEi at the base of the transformation AEi (i = 1 , 2 ,3 , 4); ej (j =1 , 2 , 3) are the corresponding unit orientation vectors along the three orthogonal axes of KEi; Lj (j = 1 , 2, 3) is the line through O with direction along ej; and Πj represents the plane passing through point O and normal to ej. The first column of the IPM indicates that the plane Π1, the joint axis L1, as well as the origin O of KE1 are invariant geometric properties of the transformation AE1, while the second column shows that the planes Π1 and Π2, as well as the line parallel to e3 passing through the origin of KE2 (one of the infinitely many line invariants of the prismatic joint) are invariant geometric properties of the transformation AE1. When an orientation vector is contained in a higher-dimensional invariant object, i.e. a plane or a line, only the higher-dimensional object invariance needs to be tracked in the IPM, as the invariance of the orientation is contained in the invariance of the higher-dimensional object. Thus, in these cases, only the higher-dimensional invariant objects are tagged with a “1” (e.g. e1 and e2 in column two are contained in Π1 and Π2, and thus only invariance properties for Π1 and Π2 are tagged in column two; while e3 is contained in line L3, and thus only the invariance property of L3 needs to be tracked). Similar conclusions can be drawn for the remaining two columns. From Eq. (5), it can be recognized that the plane Π1 of KE1 is left invariant throughout the transformations AE1, AE2, and AE3, meaning that the first three 1-DOF joints constitute a planar joint in the plane Π1 of KE1, while the origin of KE3 is invariant after the transformations AE3 and AE4 implying that the last two 1-DOF joints form a universal joint with joint center coincident with the origin of KE3 . Therefore, by using the IPM, the invariant geometric information of a serial kinematic chain can be stored in a numerical form, which is then ready to be further processed to identify the natural constraint wrenches.
α
Fig. 2. Implicit invariant geometric properties of characteristic pairs of joints.
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3. Identification of natural constraint wrenches This section presents the method for identification of natural constraint wrenches of a lower mobility serial chain featuring special geometric properties by using the IPM. By a natural constraint wrench we mean a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution. The method is developed in the light of loop closure parameters which have been used to state the geometric closure conditions for solving the inverse kinematics of serial robots [15,16]. As presented in [15], the loop closure parameters can be confined to five fundamental types expressing simple geometric relationships between points, lines or planes of a characteristic pair of joints. For a lower mobility serial chain, the proposed loop closure parameters will remain invariant due to the constraint wrenches implicitly existing in the corresponding pairs of joints while the chain is undergoing internal and/or external finite motions. These invariant geometric properties are visualized by corresponding characteristic pairs of joints as shown in Fig. 2, and the constraint wrenches identified from these characteristic pairs of joints are listed below. I) Distance between two points Fig. 2(a) illustrates a characteristic pair of a spherical joint and a universal joint having 5-DOF and requiring one constraint wrench. It can be seen that the invariant geometric property is the distance between the centers of the two joints. The corresponding constraint wrench is hence a pure force with line of action passing through the two centers of the joints. II) Distance of a point to a plane The pair of a planar joint and a universal joint forms a 5-screw system as shown in Fig. 2(b). The invariant geometric property is the distance of the center of the universal joint from any plane parallel to the planar joint plane. The constraint wrench is hence a pure force with line of action passing through the center of the universal joint and orientation normal to the plane. III) Distance of a point to a line For a pair of a spherical joint and a cylindrical joint having 5-DOF (see Fig. 2(c)), the invariant geometric property is the distance of the center of the spherical joint to the joint axis of the revolute joint. The constraint wrench is hence a pure force with line of action passing through the axis of the revolute joint and the center of the spherical joint and orientation perpendicular to the axis of the revolute joint. IV) Distance between two lines The pair of two cylindrical joints is a 4-screw system as shown in Fig. 2(d). Therefore, there are two invariant geometric properties and, correspondingly, two natural constraint wrenches. The first one is the shortest distance between the joint axes of the two revolute joints, and the corresponding natural constraint wrench is a pure force with its line of action along the common normal of the two cylindrical joint axes. V) Angle between two planes The further invariant geometric property of the pair of two cylindrical joints is the angle between two planes that are perpendicular to the joint axes (see Fig. 2(d)), respectively, resulting in a constraint couple with its direction along the cross product of the normal vectors of two planes. For quick reference, these five types of invariant geometric properties and the corresponding constraint wrenches of characteristic pairs of joints are summarized in Table 1. For a serial kinematic chain having less than f = 5 DOF, the constraint wrenches span a subspace of dimension 6 − f. In this case, the choice of 6 − f representative invariant properties and corresponding base constraint wrenches is not unique, and one may generate from any set of independent invariant properties a new equivalent set of invariant properties by suitable composition. Two typical examples are illustrated in Fig. 3. Fig. 3(a) shows a revolute-spherical dyad, which is a 4-screw system. Its invariant geometric properties could be for example chosen as the point-line distance d1 and the point-plane distance d2 to any plane normal to the revolute joint axis. Utilizing these two invariant distances, a more general invariant geometric property in terms of point-point distances can be found: as d1 and d2 are invariant, also the hypotenuse describing the distance between the center of the spherical joint and the intersection point of plane I and the revolute joint axis is invariant, and as the plane I normal to the revolute joint axis is arbitrary, any distance between the spherical joint center and a point on the revolute joint axis is invariant. Therefore, possible natural constraint wrenches are any constraint forces with line of action passing through Table 1 Five types of natural constraint wrenches of characteristic pairs of joints. Type
Invariant geometric properties
Natural constraint wrench
I II III IV V
Point–Point distance Point–Plane distance Point–Line distance Line–Line distance Plane–Plane angle
Force passing through the two points Force passing through the point with its axis normal to the plane Force passing through the line and the point with its axis perpendicular to the line Force with its axis collinear with the common normal of the two lines Couple with its direction along the cross product of the normal vectors of two planes, or of any combination of free line-direction or plane-normal orientation vectors.
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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A
Spherical joint d2
d1
I d1
II d2
Prismatic joint
II B Revolute joint
Spherical joint
I (a)
(b)
Fig. 3. Combination of invariant geometric properties of characteristic pairs.
Step 1: Find a row in the IPM that contains the longest sequence of ones in terms of joint variables contained therein. Step 2: Find a row in the IPM that contains the second-longest sequence of ones in terms of joint variables contained therein.
Step 3: Are all joints included in steps 1 & 2?
No
Yes Step 4: Combine the two rows to identify all type(s) of invariant geometric property (properties) and obtain the natural constraint wrenches of the chain by referring to Table 1. Repeat steps 1 and 2 with other independent pairs of rows until all natural constraint wrenches are found.
Full set of natural constraint wrenches has been found.
Step 5a: Keep the step-1 row fixed and proceed, one by one, with all other columns featuring a joint as follows: assume only the joint of the current column to be variable, while all other joints not contained in step-1 row are held fixed; determine for each combination of step-1 row and non-step 1 column the invariant properties subspaces by referring to Table 1.
Step 5b: Take the intersection of all invariant geometric properties to obtain the natural constraint wrenches of the chain.
Yes
Step 6: Do the obtained natural constraint wrenches span the dimension 6-f ?
No Remaining constraint wrenches must be determined numerically.
Fig. 4. Procedure to identify the reciprocal screw system of serial kinematic chains.
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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the spherical joint center and the revolute joint axis, which is the known planar pencil of constraint forces in plane II through the spherical joint center and the revolute joint axis [12]. Analogously, for a prismatic-spherical dyad shown in Fig. 3(b), the pointline distance d1 from the spherical joint center to any line parallel to the prismatic joint axis and the point-plane distance d2 from the spherical joint center to any plane containing the orientation of the prismatic joint axis are invariant. Therefore, the set of possible constraint forces forms again a planar pencil which lies in this case in plane II containing the spherical joint center and being perpendicular to the axis of the prismatic joint [12]. These cases will be further discussed in Section 4. With the aid of the IPM, the procedure to identify the natural constraint wrenches of a serial kinematic chain can be implemented by taking the following steps (see flow chart in Fig. 4 for quick reference): Step 1 Step 2 Step 3 Step 4
Find a row in the IPM that contains the longest sequence of ones in terms of the joint variables contained therein. Find a row in the IPM that contains the second-longest sequence of ones in terms of the joint variables contained therein. If the two rows obtained in steps 1 and 2 include all columns of the IPM, then go to step 4; otherwise, go to step 5a. Combine the two rows to identify the type(s) of invariant geometric property (properties) and obtain the natural constraint wrenches of the chain by referring to Table 1. Repeat steps 1 and 2 with other independent pairs of rows until all natural constraint wrenches are found. Step 5a Keep the step-1 row fixed and proceed, one by one, with all other columns featuring a joint as follows: assume only the joint of the current column to be variable, while all other joints not contained in step-1 row are held fixed; determine for each combination of step-1 row and non-step 1 column the invariant properties subspaces by referring to Table 1. Step 5b Take the intersection of all invariant properties subspaces of Step 5a to obtain the natural constraint wrench(es) of the chain. Step 6 Do the obtained natural constraint wrenches span the dimension 6-f? Yes: Full set of natural constraint wrenches has been found. No: Remaining constraint wrenches must be determined numerically.
In order to show the implementation of the proposed approach, first the example given in Section 2 will be further discussed. From the IPM given in Eq. (5), the following two combinations can be achieved by taking the steps aforementioned.
It can be found that the first one belongs to type V, Plane–Plane angle, therefore a first natural constraint wrench is a couple, while the second one is of type II, Point–Plane distance, resulting in a second natural constraint wrench of pure force. In addition, it needs to be mentioned that there is another combination that can be obtained from Eq. (5)
from which the same natural constraint wrench belonging to type II can be identified since the origins of KE3 and KE4 are coincident.
4. Examples Several typical lower mobility serial chains that are frequently used as limbs to constitute parallel mechanisms provide examples showing the application of proposed method. Here, we use R, U, P, and S to represent the revolute joint, universal joint, prismatic joint, and spherical joint, respectively. Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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Fig. 5. RRR chain.
4.1. RRR chain Fig. 5 shows a 3-DOF RRR chain, in which the joint twists $1, $2, and $3 intersect at the same point. Then the appearing transformations are AE1 ðe1 ; 0; θ1 ÞA1 AE2 ðe2 ; 0; θ2 ÞA2 AE3 ðe3 ; 0; θ3 Þ
ð6Þ
where the origin of KEi (i = 1 , 2 , 3) is defined at the intersection point, e1 of KE1, e2 of KE2, and e3 of KE3 are along the three screw axes, respectively, and A1, A2 are constant transformations involving only a rotation between adjacent joint axes, leaving the origin of KEi unchanged. Hence the corresponding IPM is
ð7Þ
from which the following three combinations of two rows of the IPM to identify the natural constraint wrenches of the chain can be obtained
which indicate that the origins of KEi (i = 1 , 2 , 3) are coincident, and the distance of the fixed origin from the plane Π1 of KE1 (Π2 of KE2 or Π3 of KE3 ) is invariant. By referring to Table 1, three forces $r1, $r2, and $r3 (type II) can be found as the natural Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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constraint wrenches as shown in Fig. 5. It has to be pointed out that other combinations of two rows of the IPM to identify natural constraint wrenches can also be achieved, as example in
Since point O is the intersection point of lines Lj (j = 1 , 2 , 3), it indicates that any forces passing through point O could be a natural constraint wrench of the chain. This conclusion is consistent with that drawn from screw theory. However, as pointline distances become singular when the distance is zero, and they are more involved to determine than point-plane distances, the latter are chosen whenever they are equivalent to the former. 4.2. UP chain A 3-DOF UP chain is illustrated in Fig. 6, in which the universal joint is decomposed into two 1-DOF revolute joints with the joint axes intersecting the axis of the prismatic joint at a common point. As shown in Fig. 6, the screw axes of $1 and $2, and the screw axis of $2 and the direction of $3 are perpendicular, respectively. Then the transformations can be given as AE1 ðe1 ; 0; θ1 ÞAE2 ðe2 ; 0; θ2 ÞAE3 ðe3 ; 1; s3 Þ
ð8Þ
where the origin of KE1 (KE2) is coincident with the center of the universal joint, while the origin of KE3 is the point of intersection of the axis of the prismatic joint and the end-effector, and e1 of KE1 , e2 of KE2 , and e3 of KE3 are along the three joint axes, respectively. Then the IPM can be formulated as
ð9Þ
Fig. 6. UP chain.
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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Therefore, we can obtain three feasible combinations
It can be seen that the origins of KE1 and KE2 as well as the planes Π2 of KE2 and Π2 of KE3 are invariant. Hence, as shown in Fig. 6, two natural constraint wrenches corresponding to the forces $r1 and $r2 (type II), and one natural constraint wrench corresponding to the couple $r3 (type V) can be identified. Note that the result remains valid if any constant rotation is inserted between AE1(e1; 0; θ1) and AE2(e2; 0; θ2). 4.3. UPR chain Fig. 7 displays a typical 4-DOF UPR chain. Compared to the UP chain (see Fig. 6), a revolute joint is placed between the prismatic joint and the end-effector, of which the screw axis of $4 is perpendicular to the direction of $3 and coplanar with the screw axis of $2. Hence the transformation related to the additional revolute joint is AE4 ðe2 ; 0; θ4 Þ
ð10Þ
where the origin of KE4 is coincident with that of KE3 at the intersection point of their joint axes, and e2 of KE4 is along the screw axis of $4. Then the IPM can be obtained from Eq. (9) by adding a fourth column.
ð11Þ
Fig. 7. UPR chain.
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Hence two feasible combinations can be achieved
from which one natural constraint wrench corresponding to the force $r1 (type II) and one natural constraint wrench corresponding to the couple $r2 (type V) as shown in Fig. 7 can be identified. Compared to the UP chain, the obtained natural constraint wrenches are nothing but two of the natural constraint wrenches of the UP chain, since by adding a revolute joint the reciprocal screw system of the UP chain is reduced and one invariant geometric property of the chain is lost. 4.4. RPS chain Fig. 8 shows a 5-DOF RPS chain, of which the spherical joint is decomposed into three 1-DOF revolute joints with the joint axes intersecting the axis of the prismatic joint at a common point. For simplicity reasons, the screw axis of $2 intersects orthogonally the screw axis of $1 at the base, while the screw axes of $3 and $4, and those of $4 and $5 are assumed to perpendicular to each other, respectively. Hence the appearing transformations are. AE1 ðe1 ; 0; θ1 ÞAE2 ðe3 ; 1; s2 ÞAE3 ðe3 ; 0; θ3 ÞAE4 ðe2 ; 0; θ4 ÞAE5 ðe1 ; 0; θ5 Þ
ð12Þ
Fig. 8. RPS chain.
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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where the origin of KE1 is coincident with the intersection point of the axes of the revolute joint and the prismatic joint, the origin of KEi (i = 2 ,3 , 4 , 5) is the center of the spherical joint, and e1 of KE1, e3 of KE2, e3 of KE3, e2 of KE4, and e1 of KE5 are along the joint axes, respectively. From Eq. (12), the IPM can be formulated as
ð13Þ
from which we can obtain.
It is easy to see that the natural constraint wrench of the RPS chain is a pure force (type II) as shown in Fig. 8 that keeps the distance of the center of the spherical joint from the plane normal to the screw axis of $1 unchanged. 4.5. General RS chain In order to illustrate the effectiveness of the proposed method in dealing with joints without orthogonal or collinear axes, four examples are further studied in this and the ensuing sections. The first example is a general 4-DOF RS chain as shown in Fig. 9, in which $i (i = 1 , 2 ,3 , 4) are coincident with the axes of 1-DOF revolute joints, respectively. The sequence of transformations can be given as A ¼ AE1 ðe1 ; 0; θ1 ÞA1 AE2 ðe1 ; 0; θ2 ÞAE3 ðe2 ; 0; θ3 ÞAE4 ðe3 ; 0; θ4 Þ
ð14Þ
Fig. 9. General RPS chain.
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where the origin of KE1 is the center of the revolute joint, while the origin of KEi (i = 2, 3 , 4) is the center of the spherical joint, e1 of KE1, e1 of KE2, e2 of KE3, and e3 of KE4 are placed along the joint axes, respectively, and A1 is a general constant transformation involving both translation and rotation between KE1 and KE2 . Then, the corresponding IPM can be formulated as
ð15Þ
Since A1 is a general constant transformations, no invariant geometric property is preserved between its base and target frames after its application. However, by combining the invariant origin of KE2 with the first column associated with the revolute joint, the following combination is obtained
from which two invariant geometric properties in terms of point-plane distance (type II) and point-line distance (type III) can be identified. It indicates that the natural constraint wrenches of this chain form a planar pencil in plane II passing through the center of the spherical joint center and the axis of the revolute joint as shown in Fig. 9. 4.6. General PS chain Another example, a general 4-DOF PS chain, is illustrated in Fig. 10. The screw axis of $1 is along the translational direction of the prismatic joint, and those of $2, $3, and $4 are coincident with the axes of 1-DOF revolute joints, respectively. Then, the sequence of transformations are A ¼ AE1 ðe1 ; 1; s1 ÞA1 AE2 ðe1 ; 0; θ2 ÞAE3 ðe2 ; 0; θ3 ÞAE4 ðe3 ; 0; θ4 Þ
ð16Þ
Fig. 10. General RSR chain.
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where the origin of KEi (i = 1 , 2 ,3 , 4) is placed at the center of the spherical joint, e1 of KE1, e1 of KE2, e2 of KE3, and e3 of KE4 are placed along the joint axes, respectively, and A1 is a general constant transformation involving only rotation between KE1 and KE2, keeping the origin of KE2 unchanged. From Eq. (16), the IPM can be written as
ð17Þ
Hence, combining the invariant origin of KE2 with the first column associated with the prismatic joint results in.
It can be seen that two invariant geometric properties of point-plane distance (type II) and one invariant geometric property of point-line distance (type III) can be identified, which indicates that the natural constraint wrenches from a planar pencil in plane II passing through the center of the spherical joint and perpendicular to the translational direction of the prismatic joint (see Fig. 10). 4.7. General RPS chain By adding a prismatic joint in the PS chain, a general 5-DOF RPS chain as shown in Fig. 11 is obtained, where the screw axis of $1 is coincident with the axis of the revolute joint. Then, the following sequence of transformations can be formulated. A ¼ AE1 ðe1 ; 0; θ1 ÞA1 AE2 ðe3 ; 1; s2 ÞA2 AE3 ðe1 ; 0; θ3 ÞAE4 ðe2 ; 0; θ4 ÞAE5 ðe3 ; 0; θ5 Þ
ð18Þ
Fig. 11. General RPS chain.
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where the origin of KE1 is the center of the revolute joint, the origin of KEi (i = 2 ,3 , 4 ,5) is placed at the center of the spherical joint, e1 of KE1 , e3 of KE2 , e1 of KE3 , e2 of KE4 , and e3 of KE5 are placed along the joint axes, respectively, and A1, A2 are general constant transformations involving rotation and translation. Hence, the corresponding IPM can be given as
ð19Þ
By combining the invariant origin of KE3 with the first and third columns associated with the revolute and prismatic joints, respectively, assuming only one of the two joints to be variable while the other is held fixed, the following IPM are obtained:
From the combination of the third row with the first and third columns, respectively, two planar pencils formed by constraint forces passing through the center of the spherical joint and lying in plane I and plane II, respectively, can be found as shown in Fig. 11. Therefore, the natural constraint wrench of this chain is a pure force passing through the center of the spherical joint and along the intersection line of plane I and plane II. 4.8. General RSR chain The fourth example with general intermediate transformations between joints is a general 5-DOF RSR chain shown in Fig. 12. Here, the screw axes of $i (i = 1 , 2 , ⋯ , 5) are grouped in three blocks, in which only the spherical joint axes intersect and the other two joint axes at the base and the tip are totally separated from the spherical joint axes. Also, intermediate constraint rotations could be interspersed between the spherical joint axis such as to allow non-orthogonal spherical joint axes, but this does not change the algorithm and is thus left out here for better visibility of the approach. Then, the sequence of transformations can be given as follows A ¼ AE1 ðe1 ; 0; θ1 ÞA1 AE2 ðe1 ; 0; θ2 ÞAE3 ðe2 ; 0; θ3 ÞAE4 ðe3 ; 0; θ4 ÞA2 AE5 ðe1 ; 0; θ5 Þ
ð20Þ
Fig. 12. General RSR chain.
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and the IPM is
ð21Þ
where A1 and A2 are general constant transformations. Analogously to the previous example, by investigating the following combinations
two planar pencils formed by constraint forces in plane I and plane II, respectively, can be identified. Consequently, the natural reciprocal screw to $i (i = 1 , 2 , ⋯ , 5) or natural constraint wrench is a pure force with line of action passing through the center of the spherical joint and along the intersection line of plane I and plane II as shown in Fig. 12.
5. Conclusions A novel and simple procedure to identify a set of natural reciprocal screws of a lower mobility serial kinematic chain with special geometric properties has been demonstrated. By a natural constraint wrench it is meant a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution. The approach starts with the definitions of the elementary transformations associated with the joints of a chain and the formulation of the invariance properties matrix that can be used to reveal the invariant geometric relationships between points, lines and/or planes of the chain. According to the types of the invariant geometric properties, the corresponding natural constraint wrenches can be classified into five groups. Then, by exploiting the invariance properties matrix, the steps to identify the invariant geometric properties and the associated constraint wrenches of a lower mobility serial chain are proposed. The procedure is demonstrated for finding the natural constraint wrenches of typical 3-, 4- and 5-DOF serial kinematic chains found in parallel mechanisms. For simplicity purposes, the examples are parameterized with simple geometric conditions, but the approach is applicable also to more general cases, as demonstrated by four non-trivial examples. While the general case requires more involved algorithms using analytical expressions, such as has been derived in the past for persistent screw systems, the proposed approach is suitable for the case of typical lower mobility chains involving special architectures of prismatic and revolute joints used in practical applications. It is believed that the present method can be extended to develop a computer-aided tool for automatic identification of the natural constraint wrenches of serial kinematic chains. Acknowledgements This research is partially supported by the National Natural Science Foundation of China (NSFC) under Grant 51405331 and the Alexander von Humboldt (AvH) Foundation of Germany. References [1] Huang T., Wang M., Yang S., Sun T., Chetwynd D. G., Xie, F., Force/motion transmissibility analysis of six degree of freedom parallel mechanisms, ASME J. Mech. Robot., 6: 031010.1 -031010.5. [2] H. Liu, T. Huang, A. Kecskeméthy, D.G. Chetwynd, A generalized approach for computing the transmission index of parallel mechanisms, Mech. Mach. Theory 74 (2014) 245–256. [3] T. Huang, H. Liu, D.G. Chetwynd, Generalized Jacobian analysis of lower mobility manipulators, Mech. Mach. Theory 46 (6) (2011) 831–844.
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[4] H. Liu, T. Huang, D.G. Chetwynd, A general approach for geometric error modeling of lower mobility parallel manipulators, ASME J. Mech. Robot. 3 (2011) 021013.1–021013.13. [5] H. Liu, T. Huang, D.G. Chetwynd, An approach for acceleration analysis of lower mobility parallel manipulators, ASME J. Mech. Robot. 3 (2011) 011013.1–011013.8. [6] L.-W. Tsai, Robot Analysis: The Mechanics of Serial and Parallel Manipulators, Wiley-Interscience Publication, 1999. [7] R.S. Ball, A Treatise on the Theory of Screws, Cambridge University Press, 1900. [8] G. Strang, Linear Algebra and Its Applications, Academic, New York, 1976. [9] J.S. Dai, J.J. Rees, Null space construction using cofactors from a screw algebra context, Proc. R. Soc. Math. Phys. Eng. Sci. 458 (2024) (2002) 1845–1866. [10] J.S. Dai, J.J. Rees, A linear algebraic procedure in obtaining reciprocal screw systems, J. Robot. Syst. 20 (7) (2003) 401–412. [11] S.A. Joshi, L.-W. Tsai, Jacobian analysis of limited-DOF parallel manipulators, ASME J. Mech. Des. 124 (2) (2002) 254–258. [12] K.H. Hunt, Kinematic Geometry of Mechanisms, Clarendon, Oxford, 1978. [13] D. Kim, W.K. Chung, Analytic formulation of reciprocal screws and its application to nonredundant robot manipulators, ASME J. Mech. Des. 125 (2003) 158–164. [14] T. Huang, S. Yang, M. Wang, T. Sun, D.G. Chetwynd, An approach to determining the unknown twist/wrench subspaces of lower mobility serial kinematic chains, ASME J. Mech. Robot. 7 (2015) 031003.1–031003.9. [15] M. Hiller, C. Woernle, The characteristic pair of joints–an effective approach for the inverse kinematic problem of robots, Proc. IEEE Int. Conf. Robot. Autom. 1988, pp. 846–851. [16] A. Kecskeméthy, M. Hiller, Automatic closed-form kinematics-solutions for recursive single-loop chains, Proc. the 22nd Biennal ASME-Mech. Conf., DE-Vol. 47 1992, pp. 387–393. [17] C.G. Gibson, K.H. Hunt, Geometry of screw systems - 1: screws: genesis and geometry, Mech. Mach. Theory 25 (1) (1990) 1–10. [18] J.M. Rico-Martínez, J. Duffy, Classification of screw systems - I. One-and two-systems, Mech. Mach. Theory 27 (4) (1992) 459–470. [19] M. Carricato, Four-dimensional persistent screw systems of the general type, Computational Kinematics, Springer, Netherlands 2014, pp. 299–306. [20] J. Angeles, Rational Kinematics, Springer Tracts in Natural Philosophy 34, Springer-Verlag, New York, 1988.
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Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance properties matrix Haitao Liu a, Andrés Kecskeméthy b,⁎, Tian Huang a,c a b c
Key Laboratory of Mechanism Theory and Equipment Design of The Ministry of Education, Tianjin University, Tianjin 300072, China Chair of Mechanics and Robotics, University of Duisburg-Essen, Duisburg 47057, Germany School of Engineering, The University of Warwick, Coventry CV4 7AL, UK
a r t i c l e
i n f o
Article history: Received 13 October 2015 Received in revised form 27 July 2016 Accepted 2 August 2016 Available online xxxx Keywords: Screw theory Reciprocal screw system Invariance properties matrix
a b s t r a c t This paper describes an automatic procedure for identifying a set of (6− f) natural constraint wrenches, termed natural reciprocal screws, for fb 6 degrees of freedom serial kinematic chains featuring special geometric properties. By a natural constraint wrench we mean a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution. This paper regards such R/P serial chains with corresponding special geometries. The approach is based on the concept of the invariance properties matrix (IPM), which was proposed in the past to find the constraint conditions for single-loop spatial mechanisms allowing for closed-form solutions. The method is further developed here to identify the invariant geometric properties associated with the natural constraint wrenches. Several typical lower-mobility serial chains are taken as examples to demonstrate the effectiveness of the approach. © 2016 Published by Elsevier Ltd.
1. Introduction Screw theory has been widely used for the kinematic and kinetostatic analyses of both serial and parallel mechanisms [1–5], and is considered to be an elegant and general mathematic tool for tackling first-order mobility and constraint problems in mechanisms. For an f (f b 6) degrees of freedom (DOF) serial kinematic chain, known as a lower mobility serial chain, the infinitesimal joint motion twists of the end-effector (“joint twists” for short) associated with its f 1-DOF joints form an f-screw system, while the constraint wrenches which are reciprocal to all possible infinitesimal joint twists of the chain form a (6− f)-screw system [6]. Since the joint twists can be obtained once the types of the joints and the configuration of the serial chain are given, the identification of the constraint wrenches becomes the main task of the concerned studies. From the concept of reciprocal screws introduced by Ball [7], if a wrench acts on a rigid body in such a way that it produces no work while the body is undergoing an infinitesimal twist, the two screws are said to be reciprocal [6]. That results in the zero ⁎ Corresponding author. E-mail address: [email protected] (A. Kecskeméthy).
http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002 0094-114X/© 2016 Published by Elsevier Ltd.
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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scalar product of any joint-twist screw $ =(S1 S2 S3 S4 S5 S6)T and the corresponding reciprocal screw $r = (Sr1 Sr2 Sr3 Sr4 Sr5 Sr6)T as $∘r ¼ 0
ð1Þ
where “∘” represents the reciprocal product of two screws. The relationship can be expanded to a set of homogeneous equations represented by the set of given linearly independent twists and its corresponding mutual unknown constraint wrenches. Hence, mathematically, the identification of constraint wrenches of a serial chain can be converted into the issue of determining the null space of a set of homogeneous equations. While the null space can be found by using different numerical methods such as Gauss Seidel elimination [8], Gram Schmidt orthogonalization [8], and an affine augmentation method [9,10], the reciprocal screws evaluated in this form are in the form of numerical values of its components, and it is not always evident how the constraint wrenches are related to geometric properties of the serial chain. Moreover, when the constraint wrenches form a subspace, an aleatory single set of independent constraint wrenches is obtained, which may not unveil clearly the intrinsic properties of the generators of the wrench subspace. An alternative approach to determine the reciprocal screws is to analyze the kinematic properties of some frequently used pairs and dyads, known as the observation method [6,11]. In particular, the reciprocal screws for all system of screws (i.e., the 1-, 2-, 3-, 4-, and 5-screw system) have been exhaustively studied by Hunt [12]. Moreover, Kim and Chung presented an analytic formulation of reciprocal screws by solving an appropriate set of algebraic equations without considering the geometry of a screw system [13]. By interpreting the reciprocal product of a wrench on a twist as a linear functional, a novel approach to determine the unknown twist/wrench subspaces of lower mobility serial kinematic chains was proposed in [14]. This allows one to automatically obtain a set of geometrically meaningful and visualizable constraint wrenches in an effective manner by taking into account three engineering constraints. Although the explicit expressions of constraint wrenches associated with a given kinematic chain can thus be achieved, these methods require a good knowledge of kinematics and screw theory, and especially they are time consuming for chains having complicated structures. Therefore, there is still a great demand for computer-aided tools for the automatic identification of the reciprocal screw system of a serial kinematic chain. For a serial lower-mobility kinematic chain, there exist in general geometric properties between adjacent or non-adjacent joints that are invariant with respect to the finite motion of the chain and/or its joints, such as intersection points, parallel and/or perpendicular conditions, and line-line distances. of several joint axes. When these invariant geometric properties are elementary, they can be defined as certain geometric measures between characteristic pairs of joints, as described in Hiller and Woernle [15], and then used to develop an effective approach for solving the inverse kinematic problem of robots. Further, by introducing the invariance properties matrix (IPM) to identify the characteristic pair of joints, Kecskeméthy and Hiller proposed a fully-automatic procedure for generating closed-form position kinematics solutions in the case of recursively solvable singleloop chains [16], where the measures between the characteristic pair of joints are interpreted as invariant characteristic projections of relative transformation matrices within branches of the loop onto scalar numbers. From the viewpoint of screw theory, the characteristic pair of joints actually indicates that constraint wrenches are invariant to motions within a serial kinematic chain. Drawing on this, this paper develops a novel and simple procedure to identify a natural reciprocal screw system, termed also a natural set of constraint wrenches, of serial lower-mobility kinematic R/P chains featuring special geometric conditions. By a natural constraint wrench we mean a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution, such as ad-hoc recognition of SE(3) submanifolds of motion of the end-effector by taking into consideration that the constraint wrenches are normal to the constraint hypersurfaces, as well as the improvement of efficiency in inverse or direct dynamics. The approach is related to the problem of finding persistent screw systems of a chain. While in the general case the task of finding persistent screw systems requires more involved algorithms using analytical expressions for each type of screw system, as developed by Gibson, Hunt, Rico, Duffy and Carricato [17–19], the proposed approach aims at providing a simple and easily applicable algorithm that can be used in cases of typical lower mobility chains with special architectures comprising only standard prismatic and/or revolute joints, such as they occur in practical applications. The rest of the paper is organized as follows: following a brief summary of the geometric properties of the projection operators involved, the formulation of IPM is described. Subsequently, the algorithm for the automatic detection of constraint wrenches of a twist system is presented. Finally, the application of the proposed approach is illustrated by typical serial kinematic chains frequently used in parallel mechanisms. While in this paper the basic procedure is illustrated on a selected set of examples, the algorithm can be generalized to cover more general cases, which may be pursued in further work. 2. Invariance properties matrix In order to define the invariance properties matrix (IPM), the algebraic and notational properties of homogeneous transformations are recollected first. Following the established convention in [20], the relative pose, i.e. rotation and translation, between two reference frames is represented by homogeneous transformation matrices which serve as transformation operators for points as well as free vectors. The general transformation law of an element “ξ” of the affine space (point or direction) is 0
ξ ¼ Aξ; A ¼
R r 0 1
ð2Þ
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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where R is the rotation matrix (actively) mapping the space at the base of the transformation to the space at its tip and transforming coordinates with respect to the tip frame to coordinates with respect to the base frame, r is the (active) translation of the origin of the base frame to the origin of the tip frame, in coordinates of the tip frame, and Rot[A] ≡R and Trans[A] ≡ r are extraction operators which can be used to select only the rotation or the translation part of the homogeneous transformation matrix A. Note that the three unit orientation vectors ei (i = 1 , 2, 3) in the direction of the coordinate axes and the position vector o representing the origin of the base coordinate system span a canonical basis of ℝ4: 0 1 1 B0C B ; e1 ¼ @ C 0A 0
0 1 0 B1C B e2 ¼ @ C ; 0A 0
0 1 0 B0C B e3 ¼ @ C ; 1A 0
0 1 0 B0C B o¼@ C 0A 1
ð3Þ
Then, the six elementary transformations can be given as follows 2
1 0 0 6 0 cθ −sθ 6 Rot½e1 ; θ ¼ 4 0 sθ cθ 0 20 0 cθ 0 sθ 6 0 1 0 Rot½e2 ; θ ¼ 6 4 −sθ 0 cθ 0 0 2 0 cθ −sθ 0 6 sθ cθ 0 Rot½e3 ; θ ¼ 6 40 0 1 0 0 0
3 0 07 7; 05 13 0 07 7; 05 13 0 07 7; 05 1
2
1 60 6 Trans½e1 ; s ¼ 4 0 20 1 60 Trans½e2 ; s ¼ 6 40 20 1 60 6 Trans½e3 ; s ¼ 4 0 0
0 1 0 0 0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0 0 0 1 0
3 s 07 7 05 13 0 s7 7 05 13 0 07 7 s5 1
where sθ (cθ) represents sinθ (cosθ). Hereafter, the elementary transformation will be denoted by AE(ei; σ; δ), translational and rotational transformations being distinguished by a Boolean variable σ (for the case of translation, its value is “1”; for the case of rotation, its value is “0”), while δ is the translational distance or rotational angle corresponding to the type of transformation. It is well-known that these elementary transformations form a basis for the group of rigid-body motions, meaning that any rigidbody motion can be theoretically decomposed into a sequence of these six transformations. By investigating the elementary transformations, it can be found that some geometric properties of the frame KE associated with AE remain unchanged after the transformation. For example, AE(e1; 0; θ) leaves the origin O of KE , the rotational axis coincident with e1, as well as the plane passing through point O and normal to e1 invariant, while AE(e2; 1; s) leaves any line along e2 as well as the two planes passing through point O and normal to e1 and e3 invariant (see Fig. 1). Hereby, it is known that the set of objects that are kept invariant by a transformation forms a group, termed the isotropy group [16]. Drawing on this property, the invariant properties related to a sequence of elementary transformations can be written in a matrix, termed the invariance properties matrix (IPM), in which the columns correspond to the individual transformations, and the rows correspond to the geometric elements which may have invariance properties. An entry is denoted by “1” if the corresponding geometric element of the row is invariant with respect to the transformation associated to the column, and otherwise by “0”. For illustration purposes, a non-rigorous example of an IPM is given below. Given a sequence of transformations A ¼ AE1 ðe1 ; 0; θ1 ÞAE2 ðe3 ; 1; s2 ÞAE3 ðe1 ; 0; θ3 ÞAE4 ðe2 ; 0; θ4 Þ
e3 Rotation
ð4Þ
1
e3
Translation
s
θ O
1
e2
O
e1
e2
e1 3 Fig. 1. Invariant properties of elementary transformations.
Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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the corresponding IPM can be formulated as
ð5Þ
where the zeros have been omitted for better clarity. Here, O is the origin of the corresponding reference frame KEi at the base of the transformation AEi (i = 1 , 2 ,3 , 4); ej (j =1 , 2 , 3) are the corresponding unit orientation vectors along the three orthogonal axes of KEi; Lj (j = 1 , 2, 3) is the line through O with direction along ej; and Πj represents the plane passing through point O and normal to ej. The first column of the IPM indicates that the plane Π1, the joint axis L1, as well as the origin O of KE1 are invariant geometric properties of the transformation AE1, while the second column shows that the planes Π1 and Π2, as well as the line parallel to e3 passing through the origin of KE2 (one of the infinitely many line invariants of the prismatic joint) are invariant geometric properties of the transformation AE1. When an orientation vector is contained in a higher-dimensional invariant object, i.e. a plane or a line, only the higher-dimensional object invariance needs to be tracked in the IPM, as the invariance of the orientation is contained in the invariance of the higher-dimensional object. Thus, in these cases, only the higher-dimensional invariant objects are tagged with a “1” (e.g. e1 and e2 in column two are contained in Π1 and Π2, and thus only invariance properties for Π1 and Π2 are tagged in column two; while e3 is contained in line L3, and thus only the invariance property of L3 needs to be tracked). Similar conclusions can be drawn for the remaining two columns. From Eq. (5), it can be recognized that the plane Π1 of KE1 is left invariant throughout the transformations AE1, AE2, and AE3, meaning that the first three 1-DOF joints constitute a planar joint in the plane Π1 of KE1, while the origin of KE3 is invariant after the transformations AE3 and AE4 implying that the last two 1-DOF joints form a universal joint with joint center coincident with the origin of KE3 . Therefore, by using the IPM, the invariant geometric information of a serial kinematic chain can be stored in a numerical form, which is then ready to be further processed to identify the natural constraint wrenches.
α
Fig. 2. Implicit invariant geometric properties of characteristic pairs of joints.
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3. Identification of natural constraint wrenches This section presents the method for identification of natural constraint wrenches of a lower mobility serial chain featuring special geometric properties by using the IPM. By a natural constraint wrench we mean a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution. The method is developed in the light of loop closure parameters which have been used to state the geometric closure conditions for solving the inverse kinematics of serial robots [15,16]. As presented in [15], the loop closure parameters can be confined to five fundamental types expressing simple geometric relationships between points, lines or planes of a characteristic pair of joints. For a lower mobility serial chain, the proposed loop closure parameters will remain invariant due to the constraint wrenches implicitly existing in the corresponding pairs of joints while the chain is undergoing internal and/or external finite motions. These invariant geometric properties are visualized by corresponding characteristic pairs of joints as shown in Fig. 2, and the constraint wrenches identified from these characteristic pairs of joints are listed below. I) Distance between two points Fig. 2(a) illustrates a characteristic pair of a spherical joint and a universal joint having 5-DOF and requiring one constraint wrench. It can be seen that the invariant geometric property is the distance between the centers of the two joints. The corresponding constraint wrench is hence a pure force with line of action passing through the two centers of the joints. II) Distance of a point to a plane The pair of a planar joint and a universal joint forms a 5-screw system as shown in Fig. 2(b). The invariant geometric property is the distance of the center of the universal joint from any plane parallel to the planar joint plane. The constraint wrench is hence a pure force with line of action passing through the center of the universal joint and orientation normal to the plane. III) Distance of a point to a line For a pair of a spherical joint and a cylindrical joint having 5-DOF (see Fig. 2(c)), the invariant geometric property is the distance of the center of the spherical joint to the joint axis of the revolute joint. The constraint wrench is hence a pure force with line of action passing through the axis of the revolute joint and the center of the spherical joint and orientation perpendicular to the axis of the revolute joint. IV) Distance between two lines The pair of two cylindrical joints is a 4-screw system as shown in Fig. 2(d). Therefore, there are two invariant geometric properties and, correspondingly, two natural constraint wrenches. The first one is the shortest distance between the joint axes of the two revolute joints, and the corresponding natural constraint wrench is a pure force with its line of action along the common normal of the two cylindrical joint axes. V) Angle between two planes The further invariant geometric property of the pair of two cylindrical joints is the angle between two planes that are perpendicular to the joint axes (see Fig. 2(d)), respectively, resulting in a constraint couple with its direction along the cross product of the normal vectors of two planes. For quick reference, these five types of invariant geometric properties and the corresponding constraint wrenches of characteristic pairs of joints are summarized in Table 1. For a serial kinematic chain having less than f = 5 DOF, the constraint wrenches span a subspace of dimension 6 − f. In this case, the choice of 6 − f representative invariant properties and corresponding base constraint wrenches is not unique, and one may generate from any set of independent invariant properties a new equivalent set of invariant properties by suitable composition. Two typical examples are illustrated in Fig. 3. Fig. 3(a) shows a revolute-spherical dyad, which is a 4-screw system. Its invariant geometric properties could be for example chosen as the point-line distance d1 and the point-plane distance d2 to any plane normal to the revolute joint axis. Utilizing these two invariant distances, a more general invariant geometric property in terms of point-point distances can be found: as d1 and d2 are invariant, also the hypotenuse describing the distance between the center of the spherical joint and the intersection point of plane I and the revolute joint axis is invariant, and as the plane I normal to the revolute joint axis is arbitrary, any distance between the spherical joint center and a point on the revolute joint axis is invariant. Therefore, possible natural constraint wrenches are any constraint forces with line of action passing through Table 1 Five types of natural constraint wrenches of characteristic pairs of joints. Type
Invariant geometric properties
Natural constraint wrench
I II III IV V
Point–Point distance Point–Plane distance Point–Line distance Line–Line distance Plane–Plane angle
Force passing through the two points Force passing through the point with its axis normal to the plane Force passing through the line and the point with its axis perpendicular to the line Force with its axis collinear with the common normal of the two lines Couple with its direction along the cross product of the normal vectors of two planes, or of any combination of free line-direction or plane-normal orientation vectors.
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A
Spherical joint d2
d1
I d1
II d2
Prismatic joint
II B Revolute joint
Spherical joint
I (a)
(b)
Fig. 3. Combination of invariant geometric properties of characteristic pairs.
Step 1: Find a row in the IPM that contains the longest sequence of ones in terms of joint variables contained therein. Step 2: Find a row in the IPM that contains the second-longest sequence of ones in terms of joint variables contained therein.
Step 3: Are all joints included in steps 1 & 2?
No
Yes Step 4: Combine the two rows to identify all type(s) of invariant geometric property (properties) and obtain the natural constraint wrenches of the chain by referring to Table 1. Repeat steps 1 and 2 with other independent pairs of rows until all natural constraint wrenches are found.
Full set of natural constraint wrenches has been found.
Step 5a: Keep the step-1 row fixed and proceed, one by one, with all other columns featuring a joint as follows: assume only the joint of the current column to be variable, while all other joints not contained in step-1 row are held fixed; determine for each combination of step-1 row and non-step 1 column the invariant properties subspaces by referring to Table 1.
Step 5b: Take the intersection of all invariant geometric properties to obtain the natural constraint wrenches of the chain.
Yes
Step 6: Do the obtained natural constraint wrenches span the dimension 6-f ?
No Remaining constraint wrenches must be determined numerically.
Fig. 4. Procedure to identify the reciprocal screw system of serial kinematic chains.
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the spherical joint center and the revolute joint axis, which is the known planar pencil of constraint forces in plane II through the spherical joint center and the revolute joint axis [12]. Analogously, for a prismatic-spherical dyad shown in Fig. 3(b), the pointline distance d1 from the spherical joint center to any line parallel to the prismatic joint axis and the point-plane distance d2 from the spherical joint center to any plane containing the orientation of the prismatic joint axis are invariant. Therefore, the set of possible constraint forces forms again a planar pencil which lies in this case in plane II containing the spherical joint center and being perpendicular to the axis of the prismatic joint [12]. These cases will be further discussed in Section 4. With the aid of the IPM, the procedure to identify the natural constraint wrenches of a serial kinematic chain can be implemented by taking the following steps (see flow chart in Fig. 4 for quick reference): Step 1 Step 2 Step 3 Step 4
Find a row in the IPM that contains the longest sequence of ones in terms of the joint variables contained therein. Find a row in the IPM that contains the second-longest sequence of ones in terms of the joint variables contained therein. If the two rows obtained in steps 1 and 2 include all columns of the IPM, then go to step 4; otherwise, go to step 5a. Combine the two rows to identify the type(s) of invariant geometric property (properties) and obtain the natural constraint wrenches of the chain by referring to Table 1. Repeat steps 1 and 2 with other independent pairs of rows until all natural constraint wrenches are found. Step 5a Keep the step-1 row fixed and proceed, one by one, with all other columns featuring a joint as follows: assume only the joint of the current column to be variable, while all other joints not contained in step-1 row are held fixed; determine for each combination of step-1 row and non-step 1 column the invariant properties subspaces by referring to Table 1. Step 5b Take the intersection of all invariant properties subspaces of Step 5a to obtain the natural constraint wrench(es) of the chain. Step 6 Do the obtained natural constraint wrenches span the dimension 6-f? Yes: Full set of natural constraint wrenches has been found. No: Remaining constraint wrenches must be determined numerically.
In order to show the implementation of the proposed approach, first the example given in Section 2 will be further discussed. From the IPM given in Eq. (5), the following two combinations can be achieved by taking the steps aforementioned.
It can be found that the first one belongs to type V, Plane–Plane angle, therefore a first natural constraint wrench is a couple, while the second one is of type II, Point–Plane distance, resulting in a second natural constraint wrench of pure force. In addition, it needs to be mentioned that there is another combination that can be obtained from Eq. (5)
from which the same natural constraint wrench belonging to type II can be identified since the origins of KE3 and KE4 are coincident.
4. Examples Several typical lower mobility serial chains that are frequently used as limbs to constitute parallel mechanisms provide examples showing the application of proposed method. Here, we use R, U, P, and S to represent the revolute joint, universal joint, prismatic joint, and spherical joint, respectively. Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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Fig. 5. RRR chain.
4.1. RRR chain Fig. 5 shows a 3-DOF RRR chain, in which the joint twists $1, $2, and $3 intersect at the same point. Then the appearing transformations are AE1 ðe1 ; 0; θ1 ÞA1 AE2 ðe2 ; 0; θ2 ÞA2 AE3 ðe3 ; 0; θ3 Þ
ð6Þ
where the origin of KEi (i = 1 , 2 , 3) is defined at the intersection point, e1 of KE1, e2 of KE2, and e3 of KE3 are along the three screw axes, respectively, and A1, A2 are constant transformations involving only a rotation between adjacent joint axes, leaving the origin of KEi unchanged. Hence the corresponding IPM is
ð7Þ
from which the following three combinations of two rows of the IPM to identify the natural constraint wrenches of the chain can be obtained
which indicate that the origins of KEi (i = 1 , 2 , 3) are coincident, and the distance of the fixed origin from the plane Π1 of KE1 (Π2 of KE2 or Π3 of KE3 ) is invariant. By referring to Table 1, three forces $r1, $r2, and $r3 (type II) can be found as the natural Please cite this article as: H. Liu, et al., An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance pr..., Mech. Mach. Theory (2016), http://dx.doi.org/10.1016/j.mechmachtheory.2016.08.002
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constraint wrenches as shown in Fig. 5. It has to be pointed out that other combinations of two rows of the IPM to identify natural constraint wrenches can also be achieved, as example in
Since point O is the intersection point of lines Lj (j = 1 , 2 , 3), it indicates that any forces passing through point O could be a natural constraint wrench of the chain. This conclusion is consistent with that drawn from screw theory. However, as pointline distances become singular when the distance is zero, and they are more involved to determine than point-plane distances, the latter are chosen whenever they are equivalent to the former. 4.2. UP chain A 3-DOF UP chain is illustrated in Fig. 6, in which the universal joint is decomposed into two 1-DOF revolute joints with the joint axes intersecting the axis of the prismatic joint at a common point. As shown in Fig. 6, the screw axes of $1 and $2, and the screw axis of $2 and the direction of $3 are perpendicular, respectively. Then the transformations can be given as AE1 ðe1 ; 0; θ1 ÞAE2 ðe2 ; 0; θ2 ÞAE3 ðe3 ; 1; s3 Þ
ð8Þ
where the origin of KE1 (KE2) is coincident with the center of the universal joint, while the origin of KE3 is the point of intersection of the axis of the prismatic joint and the end-effector, and e1 of KE1 , e2 of KE2 , and e3 of KE3 are along the three joint axes, respectively. Then the IPM can be formulated as
ð9Þ
Fig. 6. UP chain.
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Therefore, we can obtain three feasible combinations
It can be seen that the origins of KE1 and KE2 as well as the planes Π2 of KE2 and Π2 of KE3 are invariant. Hence, as shown in Fig. 6, two natural constraint wrenches corresponding to the forces $r1 and $r2 (type II), and one natural constraint wrench corresponding to the couple $r3 (type V) can be identified. Note that the result remains valid if any constant rotation is inserted between AE1(e1; 0; θ1) and AE2(e2; 0; θ2). 4.3. UPR chain Fig. 7 displays a typical 4-DOF UPR chain. Compared to the UP chain (see Fig. 6), a revolute joint is placed between the prismatic joint and the end-effector, of which the screw axis of $4 is perpendicular to the direction of $3 and coplanar with the screw axis of $2. Hence the transformation related to the additional revolute joint is AE4 ðe2 ; 0; θ4 Þ
ð10Þ
where the origin of KE4 is coincident with that of KE3 at the intersection point of their joint axes, and e2 of KE4 is along the screw axis of $4. Then the IPM can be obtained from Eq. (9) by adding a fourth column.
ð11Þ
Fig. 7. UPR chain.
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Hence two feasible combinations can be achieved
from which one natural constraint wrench corresponding to the force $r1 (type II) and one natural constraint wrench corresponding to the couple $r2 (type V) as shown in Fig. 7 can be identified. Compared to the UP chain, the obtained natural constraint wrenches are nothing but two of the natural constraint wrenches of the UP chain, since by adding a revolute joint the reciprocal screw system of the UP chain is reduced and one invariant geometric property of the chain is lost. 4.4. RPS chain Fig. 8 shows a 5-DOF RPS chain, of which the spherical joint is decomposed into three 1-DOF revolute joints with the joint axes intersecting the axis of the prismatic joint at a common point. For simplicity reasons, the screw axis of $2 intersects orthogonally the screw axis of $1 at the base, while the screw axes of $3 and $4, and those of $4 and $5 are assumed to perpendicular to each other, respectively. Hence the appearing transformations are. AE1 ðe1 ; 0; θ1 ÞAE2 ðe3 ; 1; s2 ÞAE3 ðe3 ; 0; θ3 ÞAE4 ðe2 ; 0; θ4 ÞAE5 ðe1 ; 0; θ5 Þ
ð12Þ
Fig. 8. RPS chain.
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where the origin of KE1 is coincident with the intersection point of the axes of the revolute joint and the prismatic joint, the origin of KEi (i = 2 ,3 , 4 , 5) is the center of the spherical joint, and e1 of KE1, e3 of KE2, e3 of KE3, e2 of KE4, and e1 of KE5 are along the joint axes, respectively. From Eq. (12), the IPM can be formulated as
ð13Þ
from which we can obtain.
It is easy to see that the natural constraint wrench of the RPS chain is a pure force (type II) as shown in Fig. 8 that keeps the distance of the center of the spherical joint from the plane normal to the screw axis of $1 unchanged. 4.5. General RS chain In order to illustrate the effectiveness of the proposed method in dealing with joints without orthogonal or collinear axes, four examples are further studied in this and the ensuing sections. The first example is a general 4-DOF RS chain as shown in Fig. 9, in which $i (i = 1 , 2 ,3 , 4) are coincident with the axes of 1-DOF revolute joints, respectively. The sequence of transformations can be given as A ¼ AE1 ðe1 ; 0; θ1 ÞA1 AE2 ðe1 ; 0; θ2 ÞAE3 ðe2 ; 0; θ3 ÞAE4 ðe3 ; 0; θ4 Þ
ð14Þ
Fig. 9. General RPS chain.
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where the origin of KE1 is the center of the revolute joint, while the origin of KEi (i = 2, 3 , 4) is the center of the spherical joint, e1 of KE1, e1 of KE2, e2 of KE3, and e3 of KE4 are placed along the joint axes, respectively, and A1 is a general constant transformation involving both translation and rotation between KE1 and KE2 . Then, the corresponding IPM can be formulated as
ð15Þ
Since A1 is a general constant transformations, no invariant geometric property is preserved between its base and target frames after its application. However, by combining the invariant origin of KE2 with the first column associated with the revolute joint, the following combination is obtained
from which two invariant geometric properties in terms of point-plane distance (type II) and point-line distance (type III) can be identified. It indicates that the natural constraint wrenches of this chain form a planar pencil in plane II passing through the center of the spherical joint center and the axis of the revolute joint as shown in Fig. 9. 4.6. General PS chain Another example, a general 4-DOF PS chain, is illustrated in Fig. 10. The screw axis of $1 is along the translational direction of the prismatic joint, and those of $2, $3, and $4 are coincident with the axes of 1-DOF revolute joints, respectively. Then, the sequence of transformations are A ¼ AE1 ðe1 ; 1; s1 ÞA1 AE2 ðe1 ; 0; θ2 ÞAE3 ðe2 ; 0; θ3 ÞAE4 ðe3 ; 0; θ4 Þ
ð16Þ
Fig. 10. General RSR chain.
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where the origin of KEi (i = 1 , 2 ,3 , 4) is placed at the center of the spherical joint, e1 of KE1, e1 of KE2, e2 of KE3, and e3 of KE4 are placed along the joint axes, respectively, and A1 is a general constant transformation involving only rotation between KE1 and KE2, keeping the origin of KE2 unchanged. From Eq. (16), the IPM can be written as
ð17Þ
Hence, combining the invariant origin of KE2 with the first column associated with the prismatic joint results in.
It can be seen that two invariant geometric properties of point-plane distance (type II) and one invariant geometric property of point-line distance (type III) can be identified, which indicates that the natural constraint wrenches from a planar pencil in plane II passing through the center of the spherical joint and perpendicular to the translational direction of the prismatic joint (see Fig. 10). 4.7. General RPS chain By adding a prismatic joint in the PS chain, a general 5-DOF RPS chain as shown in Fig. 11 is obtained, where the screw axis of $1 is coincident with the axis of the revolute joint. Then, the following sequence of transformations can be formulated. A ¼ AE1 ðe1 ; 0; θ1 ÞA1 AE2 ðe3 ; 1; s2 ÞA2 AE3 ðe1 ; 0; θ3 ÞAE4 ðe2 ; 0; θ4 ÞAE5 ðe3 ; 0; θ5 Þ
ð18Þ
Fig. 11. General RPS chain.
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where the origin of KE1 is the center of the revolute joint, the origin of KEi (i = 2 ,3 , 4 ,5) is placed at the center of the spherical joint, e1 of KE1 , e3 of KE2 , e1 of KE3 , e2 of KE4 , and e3 of KE5 are placed along the joint axes, respectively, and A1, A2 are general constant transformations involving rotation and translation. Hence, the corresponding IPM can be given as
ð19Þ
By combining the invariant origin of KE3 with the first and third columns associated with the revolute and prismatic joints, respectively, assuming only one of the two joints to be variable while the other is held fixed, the following IPM are obtained:
From the combination of the third row with the first and third columns, respectively, two planar pencils formed by constraint forces passing through the center of the spherical joint and lying in plane I and plane II, respectively, can be found as shown in Fig. 11. Therefore, the natural constraint wrench of this chain is a pure force passing through the center of the spherical joint and along the intersection line of plane I and plane II. 4.8. General RSR chain The fourth example with general intermediate transformations between joints is a general 5-DOF RSR chain shown in Fig. 12. Here, the screw axes of $i (i = 1 , 2 , ⋯ , 5) are grouped in three blocks, in which only the spherical joint axes intersect and the other two joint axes at the base and the tip are totally separated from the spherical joint axes. Also, intermediate constraint rotations could be interspersed between the spherical joint axis such as to allow non-orthogonal spherical joint axes, but this does not change the algorithm and is thus left out here for better visibility of the approach. Then, the sequence of transformations can be given as follows A ¼ AE1 ðe1 ; 0; θ1 ÞA1 AE2 ðe1 ; 0; θ2 ÞAE3 ðe2 ; 0; θ3 ÞAE4 ðe3 ; 0; θ4 ÞA2 AE5 ðe1 ; 0; θ5 Þ
ð20Þ
Fig. 12. General RSR chain.
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and the IPM is
ð21Þ
where A1 and A2 are general constant transformations. Analogously to the previous example, by investigating the following combinations
two planar pencils formed by constraint forces in plane I and plane II, respectively, can be identified. Consequently, the natural reciprocal screw to $i (i = 1 , 2 , ⋯ , 5) or natural constraint wrench is a pure force with line of action passing through the center of the spherical joint and along the intersection line of plane I and plane II as shown in Fig. 12.
5. Conclusions A novel and simple procedure to identify a set of natural reciprocal screws of a lower mobility serial kinematic chain with special geometric properties has been demonstrated. By a natural constraint wrench it is meant a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution. The approach starts with the definitions of the elementary transformations associated with the joints of a chain and the formulation of the invariance properties matrix that can be used to reveal the invariant geometric relationships between points, lines and/or planes of the chain. According to the types of the invariant geometric properties, the corresponding natural constraint wrenches can be classified into five groups. Then, by exploiting the invariance properties matrix, the steps to identify the invariant geometric properties and the associated constraint wrenches of a lower mobility serial chain are proposed. The procedure is demonstrated for finding the natural constraint wrenches of typical 3-, 4- and 5-DOF serial kinematic chains found in parallel mechanisms. For simplicity purposes, the examples are parameterized with simple geometric conditions, but the approach is applicable also to more general cases, as demonstrated by four non-trivial examples. While the general case requires more involved algorithms using analytical expressions, such as has been derived in the past for persistent screw systems, the proposed approach is suitable for the case of typical lower mobility chains involving special architectures of prismatic and revolute joints used in practical applications. It is believed that the present method can be extended to develop a computer-aided tool for automatic identification of the natural constraint wrenches of serial kinematic chains. Acknowledgements This research is partially supported by the National Natural Science Foundation of China (NSFC) under Grant 51405331 and the Alexander von Humboldt (AvH) Foundation of Germany. References [1] Huang T., Wang M., Yang S., Sun T., Chetwynd D. G., Xie, F., Force/motion transmissibility analysis of six degree of freedom parallel mechanisms, ASME J. Mech. Robot., 6: 031010.1 -031010.5. [2] H. Liu, T. Huang, A. Kecskeméthy, D.G. Chetwynd, A generalized approach for computing the transmission index of parallel mechanisms, Mech. Mach. Theory 74 (2014) 245–256. [3] T. Huang, H. Liu, D.G. Chetwynd, Generalized Jacobian analysis of lower mobility manipulators, Mech. Mach. Theory 46 (6) (2011) 831–844.
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