In combination with the advantages of parallel mechanisms and compliant
mechanisms, a 5-DOF compliant parallel mechanism is designed to meet the
requirements, such as large stroke, large load capacity, high precision and
high stability, for a large-aperture grating tiling device. The structure and
characteristics of the 5-DOF compliant parallel mechanism are presented. The
kinematics of the mechanism are derived based on a pseudo-rigid-body model as
well. To increase the tiling position retention stability of the mechanism, a
closed-loop control system with capacitive position sensors, which are
employed to provide feedback signals, is realized. A position and
orientation monitoring algorithm and a single neuron adaptive full closed-loop
control algorithm are proposed. Performance testing is implemented to verify
the accuracy and the tiling position retention stability of the grating
tiling device. The experimental results indicate that the tiling accuracy
reaches 0.2

Large-aperture diffraction grating is an essential element applied in many high-tech research fields, such as precision measuring instruments, optical communications and inertial confinement fusion. The currently available single-diffraction grating with a size less than 1 m cannot meet the large-aperture requirements, so it is critical to solve the problem of large-aperture diffraction grating. The tiling technique, which tiles multiple small-scale grating segments into one large grating, can be used to deal with this problem. The tiling orientation and position of two adjacent small-scale gratings should have high precision, and this precision should be maintained for a long time as well. Thus, it is important to investigate an ultra-precision positioning mechanism of high precision and high stability.

In order to achieve such high precision and stability, grating tiling theoretical analysis and structure design have respectively been studied in past decades (Bai et al., 2013; Yang et al., 2012). Zhang et al. (1998) derived the theoretical models by including various angular errors for the optical design of the array-grating compressor and analyzed the influence of angular errors on temporal performance. Kessler et al. (2004) proposed a pairing error compensation theory, which reduced the number of control variables to three. Only by controlling the longitudinal piston, angular tip and angular tilt can the grating tiling device be properly controlled. Rochester University designed a tetrahedral brace grating tiling device, OMEGA-EP, which achieved long-term tiling stability by using the interferometric near-field method (Qiao et al., 2007). The hierarchical grating tiling device was designed to meet the requirements of PETAL (Hornung et al., 2007). Habara et al. (2010) utilized three different dimensional capacitive position sensors to realize high-precision grating tiling for FIREX.

From the above literature review, it can be seen that the grating tiling
technique has become a crucial method for achieving large-aperture grating.
However, the research on grating tiling mechanism precision and stability
lags behind the development of grating tiling theory. This paper focuses on
the design of an ultra-precision positioning mechanism for a grating tiling
system. Recently, many varieties of precision positioning mechanisms have
been studied. Shi et al. (2013, 2014) and Shi and Hu (2013) presented a flexure-based hexapod
nano-positioner with small displacement, high resolution and high accuracy.
Yun et al. (2010) presented a novel 6-DOF dual-redundant compliant parallel
mechanism, and the mechanism could achieve a nanometer scale in the millimeter
range. Liang et al. (2011) also presented an ultra-precision positioning based
on compliant parallel mechanisms, which was integrated with a force sensor to
provide force feedback. Yu et al. (2015) and Hao and Yu (2016) designed an improved large-range
decoupled

Based on the above advances, a novel 5-DOF micro–nano compliant parallel mechanism with three orthogonal planes and a single-point supporting structure layout based on the actuation-leg addition method is proposed in this paper. Such a micro–nano combination design strategy increases motion stroke and improves the positioning precision. The layout improves lateral stiffness and ensures that the mechanism has high stability, large load capacity and rapid response. This paper is organized as follows: Sect. 2 presents the design and characteristics of the 5-PSS/SSP-PPS compliant parallel mechanism. Section 3 provides the kinematic analysis of the mechanism. The full closed-loop control system for high stability is described in Sect. 4. Section 5 shows the experimental system and results. Finally, conclusions are drawn in Sect. 6.

As is shown in Fig. 1, two adjacent gratings form a large-aperture grating.
In order to make the tiling errors meet the requirements, 5-DOF adjustments
should be involved between adjacent small-scale gratings: lateral shift

Schematic diagram of grating tiling.

In the design of a grating tiling device, the actuation-leg addition method is
selected as the design method because it can increase the stiffness in the
DOF directions. There are three steps in this design method. Firstly, design
an

Structure of grating tiling device.

Sketch of grating tiling device.

The spatial 5-PSS-PPS parallel structure combines with the 5-SSP-PPS parallel
structure to form a 5-DOF 5-PSS/SSP-PPS macro–micro compliant parallel
mechanism. The 5-PSS/SSP-PPS mechanism can achieve 5-DOF movements,
including 2 translational DOFs along the

The structure of unconstrained active chains is illustrated in Fig. 4. Compared with various flexure hinges, the right circular flexure hinge is considered a good candidate because of its excellent rotation capacity and lack of rotation errors (Lobontiu, 2002; Chen et al., 2008, 2009). Hence, the right circle flexure spherical joint as shown in Fig. 5 is adopted in the prototype. The double linear spring parallel combination of two linear springs (Fig. 6) is used as a flexure prismatic joint. This double linear spring not only reduces the coupling error of the single linear springs, but also increases the stiffness of the structure. And the anti-interference ability ensures that the structure has only one degree of freedom in the direction of motion, which can improve the guiding precision and working performance of the system. The PZT actuator is fixed using a pre-tightening device and a steel ball. The pre-tightening device ensures firm contact and the steel ball can help PZT actuators to withstand shear forces.

A single compliant chain.

Right circle spherical joint.

Double linear spring prismatic joint.

The grating tiling mechanism stiffness is determined by the stiffness of the flexure hinge, which plays an important role in the design of a compliant parallel mechanism. To increase the natural frequency of the mechanism so as to improve the stability and dynamic response, the design principle of the flexure hinge stiffness is that the stiffness should be as large as possible under the constraint of the stroke.

The S joint can achieve 3 rotational DOFs around three coordinate axes.
According to Castigliano's theorem (Lobontiu, 2002), the relationship of deflections

Based on the stiffness matrices method (Pham and Chen, 2005), the relationship between the
applied force

For kinematics analysis, a fixed coordinate system

Sketch of micromotion.

The coordinates of

The relationship between

Thus, the relationship between the pose of the mobile platform and input
displacement can be written as follows:

Similarly, the relationship between the pose of the mobile platform and input
displacement of macromotion can be written as follows:

The vibration in the working environment affects the stability of the grating tiling device. In order to suppress the impact of vibration and improve the stability of the grating tiling device, a full closed-loop control system, which is composed of a kinematics control model, a monitoring model and a PID control model, is designed. The basic flow of the full closed-loop control system for the stability of the grating tiling device is as follows. Firstly, the displacement changes of the capacitive position sensors are transformed into the position and angular information of the grating by using the monitoring algorithm. Secondly, it needs to be determined whether it exceeds the threshold. If it exceeds the threshold, the information is transformed into the input displacements of the actuators by using the kinematics control algorithm after it is processed by the PID control algorithm. Finally, the actuators drive the tiling mechanism to the initial position.

Schematic of single neuron adaptive PID control model.

The five degrees of freedom of two adjacent gratings can be divided into three groups and they can compensate for each other according to the compensation pair error theory of grating tiling (Kessler et al., 2004). The three groups are shift linear rotation and angular tip, lateral and longitudinal piston, and angular tilt. Thus, to achieve stability control in terms of only longitudinal piston, angular tip and angular tilt, these 3 degrees of freedom need full closed-loop control.

Capacitive position sensors merely reflect the change in linear
displacement. Therefore, it is necessary to study the monitoring algorithm
between displacement changes of the capacitive position sensors and the changes
in the position and angle of grating tiling. As shown in Fig. 3, three
capacitive position sensors F, G and H, which are distributed at the
vertices of the equilateral triangle and the connection of the capacitive
position sensors F and G is parallel to the

The relationship between the moving frame

The relative position and the orientation of the moving frame with respect to
the fixed reference frame, which can be expressed by Euler angles, are
obtained via coordinate conversion as

Thus, the Euler angles can be expressed as follows:

In order to effectively suppress the influence of vibration in the
working environment, a single neuron adaptive PID control algorithm with
self-adaptive and self-learning ability is applied to realize the
full closed-loop control as shown in Fig. 8. Single neuron self-tuning PID
control takes the ideal pose (

Out pose simulation results:

The single neuron adaptive controller adjusts the weight coefficient
according to the supervised Hebb learning rule to realize the adaptive and
self-learning function. The control algorithm and learning algorithm can be
expressed as follows:

The MATLAB program is combined with ADAMS to simulate the above single neuron PID closed-loop control algorithm. The ideal pose is configured
respectively with

In order to test the performance of the grating tiling device, experiments
for movement accuracy and stability testing are implemented.
The grating tiling system requires a nanoscale and micro-arc
level of adjustment accuracy, and vibration in the working
environment has a great impact on the experimental results. The
environment of the experiments requires constant temperature, humidity,
quietness and a high level of cleanliness in the laboratory. The experimental system
is constructed as shown in Fig. 10. It is mainly composed of a test instrument,
a grating tiling prototype and an industrial computer. As the grating is easy
damage, experiments are carried out using the same mass of aluminum
blocks instead of gratings. In this experiment, the linear
motion of the tiling mechanism was tested by using an MT2502 Heidenhain digital
length gauge and an ND 280 digital display unit with a resolution of 1

Experimental system.

In this experiment, we measure the motion accuracy of each degree in the
negative limiting position, zero point position and positive limiting
position by stepping 20 times with a 0.2

Testing results around the

Testing results around the

Testing results around the

The tables indicate that the average relative errors are below 5.5

To test the stability of the grating tiling prototype under full
closed-loop control, longitudinal piston, angular tip and angular tilt,
3-DOF motion tests, which record a set of data every
30

Stability test results of

According to a series of testing results, the grating
tiling system can achieve the anticipated accuracy of 0.2

In this paper, a novel 5-PSS/SSP-PPS compliant parallel mechanism with
a three-orthogonal-plane and single-point supporting structure is proposed.
The kinematics are derived based on a pseudo-rigid-body model. To achieve high
stability, a full closed-loop control system a including monitoring
model and a PID control model is proposed. The accuracy and stability testing
results show that the step accuracy reaches 0.2

All the data used in this paper can be obtained upon request from the corresponding author.

The authors declare that they have no conflict of interest.

This work is partially supported by the National Natural Science Foundation of China (grant no. 51475114) and the Natural Science Foundation of Hei Long Jiang Province (grant no. E201424). Edited by: Guangbo Hao Reviewed by: Shixun Fan and one anonymous referee