Geometric construction and kinematic analysis of a 6R single-loop overconstrained spatial mechanism that has three pairs of revolute joints with intersecting joint axes

This paper deals with the geometric construction and kinematic analysis of a single degree-of-freedom 6R overconstrained spatial mechanism (6R mechanism for short) that has three pairs of revolute (R) joints with intersecting joint axes. How to construct a 6R mechanism with three pairs of R joints with intersecting joint axes is presented ﬁrst. The kinematic analysis of the 6R mechanism is then discussed. The analysis shows that the 6R mechanism usually has two solutions to the kinematic analysis for a given input. In two conﬁgurations in each circuit of the 6R mechanism, the axes of four R joints are coplanar, and the axes of the other two R joints are perpendicular to the plane deﬁned by the above four R joints.

Although the successful industrial applications of single-loop overconstrained mechanisms are not many so far, single-loop overconstrained mechanisms have been used in the development of deployable structures [14] and parallel mechanisms [27,28].The application of 6R mechanisms in mobile robots [29] and other devices [25,30] is being explored.
Recently, several 6R mechanisms, which have two pairs of adjacent R joints with parallel joint axes and one pair of non-adjacent R joints with parallel joint axes, have been systematically obtained using a dual-quaternion based approach [19,20,21] or a construction approach [22,24].Before the systematic type synthesis of this class of 6R mechanisms, only two such 6R mechanisms were proposed in the literature and both were unfortunately ignored by researchers working on overconstrained 6R mechanisms for decades [11,18,19,22].The first one was proposed by Mudrov in 1976, as pointed out in [25], and can be regarded as a special case of the double-Bennett hybrid 6R mechanism [4].The second one was obtained during the investigation of novel steering techniques in reference [9].Following the success of type synthesis of this class of 6R mechanisms, it is logic to obtain 6R mechanisms that have three pairs of R joints with intersecting joint axes, including two pairs of adjacent R joints with intersecting joint axes and one pair of nonadjacent R joints with intersecting joint axes.It is apparent that two 6R mechanisms that have three pairs of R joints with intersecting joint axes can be obtained from plane symmetric Bricard mechanism and line symmetric Bricard mechanism by imposing certain geometric constraints, like in the generation of 6R mechanisms that have three pairs of R joints with parallel joint axes [22].Whether there are other classes of 6R mechanisms that have three pairs of R joints with intersecting joint axes has not been investigated.
Inspired by the geometric construction of Bricard 6R mechanisms [1,2], especially the trihedral Bricard mechanism and the Type III Bricard's flexible Octahedron, this paper aims at revealing a 6R mechanism that has three pairs of R joints with intersecting joint axes using a geometric construction approach.
This paper is organized as follows.The geometric construction of a 6R mechanism that has three pairs of R joints with intersecting joint axes will be given in Section 2. The kinematic analysis of the 6R mechanism will be presented in Section 3. In Section 4, the characteristics of the 6R mechanism will be revealed.Finally, conclusions will be drawn.
For simplicity reasons, sin θ i and cos θ i are denoted by Sθ i and Cθ i , respectively.

Description of a 6R mechanism that has three pairs of R joints with intersecting joint axes
In this section, the geometric construction of a 6R mechanism is presented first.The D-H link parameters of the 6R mechanism are then given.

Geometric construction of a 6R mechanism
A 6R mechanism that has three pairs of R joints with intersecting joint axes can be constructed as follows (Fig. 1): • Lines AA ′ and BB ′ that are perpendicular to the kite ACBD.
• Lines CA 1 and CB 1 that are tangent to circles A and B respectively at points A 1 and B 1 such that ∠ACA 1 = ∠BCB 1 .
• Lines DA 2 and DB 2 that are tangent to circles A and B respectively at points A 2 and B 2 such that ∠ADA 2 = ∠BDB 2 .
Lines AA ′ , BB ′ , CA 1 , CB 1 , DA 2 and DB 2 are the six lines required for placing six R joints.
Step 2: Construct a 6R mechanism using the six lines obtained in Step 1.
Place six R joints 1, 2, and CB 1 respectively and connect them in the sequence of 1-2-3-4-5-6-1.One then obtains a 6R mechanism that has three pairs of R joints with intersecting joint axes 1-2-3-4-5-6-1.Here two adjacent R joints are connected using a circular link if the two joint axes intersect with each other or a straight link if the the two joint axes do not intersect.
From Fig. 1, We obtain It is noted that Eq. ( 1) then becomes

Link parameters of the 6R mechanism
Different variations of D-H notations are used in the literature [8,12,14,31,32,33].The coordinate frames are attached to the links and the link parameters are defined as in [33] (Fig. 2).Z i -axis is along the axis of joint i.X i -axis is along the common perpendicular between Z i−1 -and Z i -axes.O i is the intersection of X i -and Z i -axes.Y i -axis is defined by X i -and Z iaxes through the right handed rule.The link parameters of link i are: d i (the distance between X i -and X i+1 -axes measured from X i -axis to X i+1 -axis along Z i -axis), α i (the twist angle between Z i -and Z i+1 -axes measured from Z i -axis to Z i+1 -axis about X i+1 -axis), and l i (the distance between Z i -and Z i+1 -axes measured from Z i -axis to Z i+1 -axis along X i+1 -axis).Throughout this paper, link 6, which connects joints 6 and 1, is selected as the frame of the 6R mechanism and highlighted in blue.
The link parameters of the 6R mechanism are: From the construction of the 6R mechanism (Fig. 1), we learn that the link parameters of the 6R mechanism satisfy the following constraint (see Eq. ( 2)) : As pointed out by one of the reviewers of this paper, this 6R mechanism that has three pairs of R joints with intersecting joint axes is a special case of the Wohlhart 6R mechanism [8] obtained by merging two Goldberg 5R mechanisms.The contribution in this paper is that this 6R mechanism is obtained by a concise geometric construction.If the kite for constructing the 6R mechanism (Fig. 1) is turned into a rhombus, then the 6R mechanism that has three pairs of R joints with intersecting joint axes is a special case of a Bricard line symmetric 6R mechanism.
The variation of θ i (i=2, 3, . . .6) with respect to θ 1 for Mechanism I is shown in Fig. 3.It can be readily verified using Eq. ( 6) that the axes of joints 2 and 5 are always coplanar during the motion of the 6R mechanism.It is observed that Mechanism I has two circuits with joints 1, 6, 3 and 4 as full-turn R joints.Figures 4 and 5 show Mechanism I at configurations A, B and C in circuit 1 and configurations D, E and F in circuit 2 respectively.It is observed that at two configurations in each circuit (see configurations A and C in circuit 1 and configurations D and F in circuit 2), the axes of R joints 1, 3, 4 and 6 of Mechanism I are coplanar and the axes of R joints 2 and 5 are perpendicular to the plane defined by the axes of R joints 1, 3, 4 and 6.

Discussion
The above results have been verified using several mechanism models built using 3D printing.Figure 6 shows the CAD model and 3D-printed prototype of 6R Mechanism I (Fig. 4c).It is noted that joints 1 and 6 in this prototype are prevented from full-cycle rotation due to interference between links 2 and 4 as well as links 1 and 5.
Let K 1 , K 2 and K 3 denote the intersections of joint axes of joints 1 and 6, joints 2 and 5, and joints 3 and 4. Π 1 , Π 2 and Π 3 represent the plane defined by the axes of joints 1 and 6, joints 2 and 5, and joints 3 and 4 respectively (Fig. 7).From [36], we obtain that planes Π 1 ,Π 2 and Π 3 and plane K 1 K 2 K 3 have a common point, K, at any configuration of the 6R mechanism during motion.It is noted that at configurations A, C (Fig. 4), D and F (Fig. 5) of Mechanism I, point K 2 is at infinity.

Conclusions
A 6R mechanism that has three pairs of R joints with intersecting joint axes has been proposed using a geometric construction approach.Kinematic analysis of the mechanism has been presented.The analysis has shown that the 6R mechanism usually has two solutions to the kinematic analysis for a given input.In two configurations in each circuit of the 6R mechanism, the   axes of four R joints are coplanar, and the axes of the other two R joints are perpendicular to the plane defined by the above four R joints.
Although the 6R mechanism that has three pairs of R joints with intersecting joint axes proposed in this paper is a special case of Wohlhart's 6R mechanism based on two Goldberg 5R mechanisms, it is obtained here using a very concise geometric construction.The kinematic analysis of the 6R mechanism in this paper is also more concise than that in reference [8], where (a) CAD model.the kinematic analysis was based on the kinematic analysis of two Goldberg 5R mechanisms.
This work enriches the geometric approach for identifying 6R overconstrained mechanisms and is a step forward in the research on 6R mechanisms that have three pairs of R joints with intersecting joint axes.

Figure 2 :
Figure 2: D-H parameters of the 6R mechanism.

Figure 6 :
Figure 6: A prototype of 6R Mechanism I.