Since precise linear actuators of a compliant parallel manipulator suffer from their inability to tolerate the transverse motion/load in the multi-axis motion, actuation isolation should be considered in the compliant manipulator design to eliminate the transverse motion at the point of actuation. This paper presents an effective design method for constructing compliant parallel manipulators with actuation isolation, by adding the same number of actuation legs as the number of the DOF (degree of freedom) of the original mechanism. The method is demonstrated by two design case studies, one of which is quantitatively studied by analytical modelling. The modelling results confirm possible inherent issues of the proposed structure design method such as increased primary stiffness, introduced extra parasitic motions and cross-axis coupling motions.
Compliant/flexure parallel manipulators (CPMs) have experienced rapid development over the two past decades due to their merits of eliminated backlash, friction, wear and lubrication as well as a reduced number of parts, which are very suitable for applications in precision engineering (macro scale) and MEMS (micro scale) (Smith, 2003; Howell et al., 2013). When high-precision motion is required, contact linear actuators such as piezoelectric (PZT) (Yue et al., 2010) and voice coil (VC) (Hiemstra et al., 2012) are the options because of their excellent positioning performances. However, these linear actuators conventionally suffer from their inability to tolerate the transverse motions/loads in the multi-axis motion requirements. Therefore, actuation isolation that is defined as eliminating the transverse motion at the point of actuation (Awtar and Slocum, 2007) should be considered in the CPM design. In compliant mechanisms, completely eliminating the transverse motion and allowing the actuation point to perfectly move along the actuation direction is impossible due to the nature of elasticity. Therefore, we aim to minimise the transverse motions of the linear actuator in the multi-axis motion system, by using a kind of decoupling mechanism that can transmit actuation force with negligible motion loss and can accommodate/absorb transverse motions without producing transverse loads to the actuator (Awtar and Slocum, 2007).
There are a few successful methods to design CPMs, which include
constraint-based method (Hale, 1999), FACT (Hopkins and Culpepper, 2010), and screw-theory-based method (Su et al., 2009).
Those approaches mainly focus on design compliant
Based on the above advances, this paper aims to propose a general method for the conceptual design of CPMs with actuation isolation, which intends to complement the existing methods. The remainder of this paper is organised as follows. Section 2 presents a general conceptual design approach by adding actuation legs. Case studies using the proposed method to design two CPMs with actuation isolation are discussed in Sect. 3. Analytical modelling is implemented in Sect. 4 to show the quantitative characteristic analysis of one resulting CPM. Section 5 draws the conclusions.
The design approach for actuation isolation in this section is a
constraint-based design method, which is based on the following rule of
rigid-link or compliant parallel mechanisms (see Kong and Gosselin, 2007 for instance):
The detailed procedure to design a CPM with actuation isolation is given below:
Design an Design the same number of extra compliant legs as the number of DOF
specified in Step 1, where each leg is a simple 6-DOF serial mechanism that
consists of a 1-DOF translational (or prismatic) joint and a 5-DOF straight
wire beam with its axis along with the direction of the translational joint. Connect the wire beam in each extra leg to the motion stage of the original
mechanism to comply with the following conditions for arranging each leg:
the connection point between the wire beam in one extra leg and the motion
stage should be able to be actuated by the linear actuator guided by the
translational joint; and Appropriately arrange the configuration according to specific requirements.
Note that the wire beam (or other alternative design) in each extra leg
offers the decoupling of the transverse motions on the actuation (Awtar et
al., 2012; Hopkins et al., 2012) where the actuator is guided by a translational joint. The length of the
wire beam in each added leg can be adjusted for specific reasons such as
reducing the number of the DOF of the actuation leg; and the translational
joint can have different forms even though with amplification or
direction-changing mechanisms. The above constraint-based design method can
be seen as an “actuation-leg addition” method, which can be used to design
any-DOF CPMs that are dominated by the original compliant mechanisms.
This section designs two 3-DOF CPMs for demonstrating the proposed actuation leg addition method: one is the in-plane 3-DOF 2T1R (T: translational; R: rotational) CPM (Fig. 1), and the other is the out-of-plane 3-DOF 2T1R CPM (Fig. 2).
Design of an in-plane 3-DOF 2R1T CPM with actuation isolation.
Design of an out-of-plane 3-DOF 2R1T CPM with actuation isolation.
The design of the in-plane 3-DOF CPM starts from designing an original in-plane 3-DOF compliant mechanism without considering actuation isolation (Fig. 1a). This original planar mechanism consists of 4 identical parallel wire beams with no intersection point (intersection at infinitely far position), which can be designed using the emerging method (Hopkins and Culpepper, 2010; Su et al., 2009). By adding three extra 6-DOF legs (Fig. 1b) where the guiding translational joint is a leaf-type parallelogram mechanism, an in-plane 3-DOF CPM with actuation isolation is obtained as shown in Fig. 1c. The FEA (finite element analysis) motion demonstrations are illustrated in Fig. 1d–f, which show three individual motions in the DOF directions.
Designing the out-of-plane 3-DOF CPM also starts from designing an original out-of-plane 3-DOF mechanism without taking actuation isolation into account (Fig. 2a). This original out-of-plane mechanism is a symmetrical design for better performances. It is composed of 6 identical parallel wire beams in the same plane with more than one intersection points, where the distance between any two intersection points are large enough. This original mechanism was reported in Hao (2016), which can actually be seen as a type of well-known 6-DOF Gough-Stewart platform at its planar actuation singularity position. By adding three extra 6-DOF legs (Fig. 1b), an out-of-plane 3-DOF CPM with actuation isolation is obtained as shown in Fig. 2b. The FEA motion demonstrations are illustrated in Fig. 2c and d.
Geometry and coordinate system representations.
Although the proposed design method in Sect. 2 can effectively address the issue of actuation isolation, it may be only valid with certain tolerances, i.e., it may introduce other problems such as parasitic motions and amplified actuation stiffness. In order to quantitatively assess the performance characteristics of the resulting CPM with actuation isolation compared to its original mechanism, analytical modelling and compliance analysis with regard to the output motion stage can be carried out (Hao, 2016; Hao and Kong, 2013). It should be clarified that the motion range consideration of the modelled example based on the strength theory is out of scope of this paper.
We take the design in Fig. 1 with a specific legs' arrangement as an example for comprehensive analysis in
this section, whose all geometry and coordinate system representations are
clearly detailed in Fig. 3. All beams are assumed to have the same
Young's modules (
The stiffness matrix for the original mechanism with regard to the centre of
the motion stage is to be developed at first based on the work in Hao and Kong (2013). The
stiffness matrix of each wire beam with square cross sections in Fig. 3c
is expressed as
The stiffness matrix of the original mechanism can be then derived based on
the rule of parallel systems as follows (Hao and Kong, 2013):
The compliance matrix of the original mechanism can be produced as
The stiffness matrix of a leaf beam in the 1-DOF translational joint can be
obtained as below:
The stiffness matrix of the translational joint composed of two identical
leaf beams can be derived as follows (Hao, 2016):
Therefore, the compliance matrix of the extra leg can be derived based on
the rule of serial systems (Hao and Kong, 2013) as:
Finally, the stiffness matrix, with regard to the global coordinate system
The compliance matrix of the CPM with actuation isolation can be thus obtained as
However, in comparison with the original mechanism the added legs with the current specific arrangement can result
in two problems as below, exposing trade-offs to be considered:
The extra legs do increase the stiffness in the DOF directions, which is
reflected by the decreased primary compliances in Eq. (10b). For the
analysis case study in this section, the primary compliances of the CPM with
actuation isolation are reduced by 29.67 % in the The added legs can cause issues associated with parasitic motions (related
to DOC) and cross-axis coupling (related to DOF) that do not exist in the
original mechanism. For instance, the force, applied on the origin of the
global coordinate system along the
Note that the resulting CPM with actuation isolation has alleviated existing
parasitic rotation in the
The above analytical modelling method can also be used to quantitatively analyse the stiffness/compliance of the proposed out-of-plane 3-DOF CPM with actuation isolation (Fig. 2), which is left for the future work. The modelling results would be useful to identify if the added legs are able to actuate all DOF directions and/or if there exists decoupling between input and output. If each primary compliance in the compliance matrix of the resulting CPM includes the contribution of at least one of the actuation compliances of the translational joints in added legs, all DOF are actively controllable. It should be mentioned that a 6-DOF CPM with actuation isolation can be proposed using the method presented in Sect. 2 where the original compliant mechanism is a 0-DOC (6-DOF) mechanism (free rigid body). The 6-DOF CPM is composed of six actuation legs, each of which serially consists of a 1-DOF translational joint and a 5-DOF wire beam with its axis along with the direction of the translational joint. For practical reasons, the six wire beams are arranged in three pairs along three orthogonal directions (Hale, 1999).
An actuation-leg addition approach has been presented in this paper for the conceptual design of any-DOF CPMs with actuation isolation. The added leg is a 6-DOF serial chain, which is a very simple design in compliant mechanisms and can be arranged in different desired positions. This method provides an universal/general solution to consider actuation isolation for any (compliant) parallel mechanisms. Two design studies have been implemented with one case analytically modelled. The linear modelling results quantitatively show that the added extra legs, although obtaining the actuation isolation, may cause several inherent problems including increased primary stiffness, introduced extra parasitic motions and cross-axis coupling motions.
It is hoped that the proposed method can be used as an initial solution for further comprehensive modelling and geometrical optimization of CPMs towards specific applications. Trade-offs should be carefully considered between the strength and weakness using actuation isolation. Further experimental verification is also to be investigated.