Abstract
The effects on deployment accuracy of joint clearances and forces acting on the deployable mechanism are investigated. “A virtual massless link” is used to describe the joint clearance of the mechanism in the free state. Aiming at the following problems of multiloop linkages mechanical system such as loop constraints coupling and complicated topology structure, the geometric relationships and the deploying error equations of deployable mechanism are provided based on DenavitHartenberg matrix by splitting multiloop linkage into 3 singleloop linkages. In addition, with consideration of forces, “a virtual massless rope” is used instead of “virtual massless link” to describe the joint clearance for avoiding the extra singularities. And the criterion of mechanism in stable configuration is obtained by weighting coefficient method on the basis of minimum potential principle. The solving procedure of deploying error equations in the state of forces constrained is presented by combining MonteCarlo method with genetic algorithm, it is an effective solution for the nonlinear objective function with multiple constraints and strong coupling. The results indicate that the appropriate forces acting on mechanism produce a significant effect on improving repeat accuracy, and the absolute accuracy can be improved with the future error compensation.
Highlights
 The effects on deployment accuracy of joint clearances and force are investigated.
 A virtual massless rope is used to describe the joint clearance for avoiding the singularities.
 The criterion of mechanism in stable configuration is obtained.
 The MonteCarlo method and genetic algorithm are combined for solving and analyzing the deployment error.
1. Introduction
Deployable mechanism has been widely applied in the supporting device of space equipment such as flexible solar panels, satellite antenna or space telescope for the characteristics of big stowed volume ratio, light weight, high structural accuracy, etc. And the deployment accuracy is the key to ensure the normal operation [14]. Common deployable mechanisms are constructed with hinged bars as well as with the tightrope connected for enhancing stability [58]. In practice, the joint clearance is inevitable due to manufacture errors and the need for superior rotation capability. It performs a kinematic uncertainty on mechanism, and affects the deployment accuracy [914].
Plenty of researchers have been devoted to studying the influence of joint clearance. Most of the studies concentrated on singleloop linkage [1520], but the common deployable mechanisms are multiloop linkages that have loop constraints coupling and complicated topology structure [21, 22]. The analytical method used in singleloop linkage may not be applicable to it on account of challenges of highly nonlinear and strong coupling. Few achievements have been obtained on accuracy analysis of multiloop linkage with joint clearances to be considered, Yang Y et al. [22] have analysed the relationships between distortion of whole mechanism and deviation of each rod based on the minimum of elastic deformation energy and Lagrange multiplier method, but the influence of joint clearance has not been considered. Ding J. and Wang C. [23] have decoupled the multiloop mechanism with joint clearances based on matrix method and particle swarm optimization, but they only have investigated the maximum deployment error. Tsai M. J and Lai T. H. [24] provided a generalized method for error analysis of multiloop mechanism with joint clearance on the base of screw theory, however, the effort to diminish the influence of joint clearance on the mechanism accuracy has not involved. Zhang X. and Zhang X. [25] have minimized the influence of joint clearance by controlling the redundant actuations of mechanism of which the redundant degree of freedoms and actuations are needed, and result in complex controlling and configuration change of mechanism.
Furthermore, the solution procedure of error involves an optimization problem of the nonlinear objective function with multiple constraints and strong coupling. So, it is difficult to be solved by traditional solutions. Heuristic algorithm is an important means to solve this problem, but it is still in the research phase. The classical algorithms have a lot of probability with the deficiencies of solving inefficiency, poor robustness and easy to fall into local optimal solution. It is usually necessary to improve the algorithm according to the actual tasks, or even a need of several algorithms collaboration to handle complex tasks. For instance, to improve the efficiency, Liu Q. et al. [26] proposed a speculative approach to spatialtemporal efficiency with multiobjective optimization by using Kernelbased Extreme Learning Machine algorithm and TSNSGAⅡ. Kong Y. et al. [27] proposed a belief propagationbased method. Deng W et al. [28] investigated an improved adaptive particle swarm optimization(DOADAPO) algorithm based on making full use of the advantages of Alphastable distribution and dynamic fractional calculus. Rong H. et al. [29] proposed a novel K^{+}isomorphism method and an improved MPDV method. In addition, in order to accomplish the complex tasks, some current techniques are combined with the K^{+}isomorphism method. And in order to avoid the algorithm to get into the local optimal solution as well as to improve the robustness and the solution accuracy of the algorithm, Gu B. and Sheng V. S. [30] proposed a robust regularization path algorithm based on a new equivalent dual formulation for $\nu $SVC and a robust νSvcPath. Yu Xue et al. [31] introduced a selfadaptive mechanism of the artificial bee colony algorithm. Deng W. et al. [32] proposed a genetic and ant colony adaptive collaborative optimization (MGACACO) algorithm by introducing the chaotic optimization method, multipopulation collaborative strategy and adaptive control parameters into the GA and ACO algorithm.
The object of this study is to propose an accuracy analysis method for multiloop mechanism with joint clearance and to provide an approach for improving the deployment accuracy without any redundant actuation added. First, the equivalent model of deployable mechanism with joint clearance is obtained by converting the joint clearance into “a virtual massless link”. And in order to decouple the multiloop, the geometrical relationship is expressed based on DenavitHartenberg (DH) theory, while to split the multiloop linkage into some singleloop linkages that subject to the same constraints. Besides, the deployment error is analysed by simulating the manufacture errors and the state of joint clearance without any constraint on deployable mechanism (the free state) with the use of MonteCarlo method. In addition, the influence of the forces acting on the deployable mechanism is researched for diminishing the deployment error. “A virtual massless rope” is used instead of “a virtual massless link” for modelling the deployable mechanism with joint clearance and forces constrained. The criterion of mechanism in stable configuration is obtained by weighting coefficient method on the basis of minimum potential principle. And according to the characteristics of deployment error model, the MonteCarlo method and genetic algorithm are combined to solve and analyse the deployment error with forces constraint. Furthermore, this research provides a reference for error correction and system reliability analysis of mechanism.
2. Configurations of cablestrut deployable mechanism
This study is based on the antenna supporting mechanism of the Engineering Test Satellite Ⅷ(ETSⅧ) for the research [33]. The supporting mechanism consists of 14 basic modules, and every basic module contains 6 basic frames. In addition, the adjacent basic frames are connected by cables for improving the rigidity and stability, as shown in Fig. 1.
The basic frame is the bases of the antenna supporting mechanism to get the capabilities of deployment and keeping deployment accuracy. Fig. 2 shows the working principle. The slider A moves along the y direction under the action of spring, and drives the hinged bars to motion until the link DE reached the mechanical limit point H where the link DE and QD are collinear and cables are tight. So, the mechanism is selflocked and subjected to tension and spring load while in deployed state. M1 and M2 in Fig. 2 are the fixed positions of support cables [33] that the positional accuracy in deployed state operating a directly effect on surface accuracy of antenna.
Fig. 1Basic module of ETSⅧ
a) Deployed state
b) Folded state
Fig. 2Basic frame of ETSⅧ
a) Folded state
b) Deployed state
3. Model of cablestrut deployable mechanism with joint clearances
The basic frame of ETSⅧ is a single degree of freedom mechanism that contains one sliding pair and 9 revolute joints. The clearance exists in any motion pair for motion needed and manufacturing error. In order to improve reliability, the linear bearing is used for precisely fitting together with axis in sliding pair, the clearance is minimal and can be neglected. So, the model of basic frame can be simplified into a mechanism only with joint pairs.
3.1. Equivalent model of joint clearance
The basic frame pose of ETSⅧ will no longer change as it reached the limit position where every link is in the stable state of force balance, so the revolute joints are met to the contact hypothesis that the hole and shaft can be regarded as always stay in touch. And the basic frame can be treated as a multirigidbody system for the less stress and shock. Therefor the joint clearance can be modelled as “a virtual massless link” [34], as shown in Fig. 3, the eccentricity of the hole and shaft ${r}_{i}$ is described as “a virtual massless link” that both ends are respectively hinged with the centres of the hole and shaft.
Fig. 3The model of joint clearance
${r}_{i}$ is a normaldistributed random variable which determined by the tolerance of hole and shaft. According to the 3 $\sigma $ principles, almost all values of ${r}_{i}$ can be obtained in the interval ($\mu 3\sigma $, $\mu +3\sigma $), the mean value ${\mu}_{i}$ and standard deviation ${\sigma}_{i}$ of ${r}_{i}$ are known by Eq. (1):
where $\mathrm{max}\left({r}_{i}\right)$ and $\mathrm{min}\left({r}_{i}\right)$ denote respectively the maximum and minimum of clearance.
Building the reference frames and orienting the positive direction of azimuth angles ${\theta}_{i}$, ${\theta}_{j}$ of links based on DH theory. Where ${\theta}_{i}$ is the azimuth angle of “virtual massless link” ${O}_{i}{{O}_{i}}^{\text{'}}$ relative to link $i$, ${\theta}_{j}$ is the azimuth angle of link $j$ relative to “virtual massless link” ${O}_{i}{{O}_{i}}^{\text{'}}$, and meet the relationship of Eq. (2):
where $\alpha $ is the azimuth angle of link $j$ relative to link $i$ which are linked by revolute pair with joint clearance.
3.2. Deployment error model in the free state
While regardless the influence of forces such as cabletension, spring force, etc., the relative position between the hole and shaft of the revolute pair is conform to random, i.e., while the “virtual massless link” in a free state, its azimuth angle ${\theta}_{i}$ obey the uniform random distribution in [$\pi $, $\pi $), and the ${\theta}_{i}$ is a constant value at any certain moment. Fig. 4 shows the model of basic frame with joint clearance that based on the “virtual massless link”, and it only has one degree of freedom for the “virtual massless links” are stationary at any certain moment. For the joint clearance at D point, there is an angle $\beta $ between link DE and link ${Q}^{\text{'}}D$ after basic frame into the deployed state, as shown in Fig. 5.
According to sine theorem, the relationship can be expressed as follow:
By Eq. (3):
According to the definition of DH theory:
where ${\theta}_{D}$ and ${\theta}_{{D}^{\text{'}}}$ are the azimuth angles of link $D{D}^{\text{'}}$ relative to link ${Q}^{\text{'}}D$ and link ${D}^{\text{'}}H$ relative to link $D{D}^{\text{'}}$, respectively; ${r}_{D}$ is the eccentricity of the hole and shaft at hinged point $D$, ${l}_{DH}$ is the distance of point $D$ and $H$.
Fig. 4The model of basic frame with joint clearance in free sate
Fig. 5The model of revolute joint D with clearance
For easy to analysis, splitting the basic frame model (Fig. 4) into three singleloop models of Ⅰ, Ⅱ, Ⅲ, as shown in Fig. 6 which subject to the same constraints to model in Fig. 4.
Fig. 6The structure of singleloop
a) Loop Ⅰ
b) Loop Ⅱ
c) Loop Ⅲ
Let $i,j\in \{O,P,{P}^{\text{'}},Q,{Q}^{\text{'}},R,{R}^{\text{'}},B,{B}^{\text{'}},C,{C}^{\text{'}},D,{D}^{\text{'}},E,{E}^{\text{'}},F,{F}^{\text{'}},G,{G}^{\text{'}},M2\}$, and ${{}_{}{}^{j}\mathbf{T}}_{i}^{}$ be the conversion matrix of reference frame ${{o}_{j}x}_{j}{y}_{j}{z}_{j}$ to ${{o}_{i}x}_{i}{y}_{i}{z}_{i}$. Attaching moving frame ${{o}_{i}x}_{i}{y}_{i}{z}_{i}$ to the revolute joints of loop Ⅰ, Ⅱ, Ⅲ based on DH theory, so the general conversion matrix between two reference frames can be given as follow:
where:
${l}_{j}$ is the measured distance from ${o}_{j}$ to ${o}_{i}$ along ${x}_{j}$, ${\omega}_{j}$ is the rotation angle from ${z}_{j}$ to ${z}_{i}$ around ${x}_{j}$, ${d}_{i}$ is the measured distance from ${x}_{j}$ to ${x}_{i}$ along ${z}_{i}$, ${\theta}_{i}$ is the rotation angle from ${x}_{j}$ to ${x}_{i}$ around ${z}_{i}$.
Therefore, the relationships in the fixed reference frame $o$$xyz$ of points on basic frame can be expressed as follows:
Loop Ⅰ:
Loop Ⅱ:
Loop Ⅲ:
The basic frame is a multiclosed loop, and the clearance in slider A is ignored, so the following constraints should be met:
Besides, for the clearance of revolute joint, the azimuth angle of link $j$ relative to link $i$ has an additional angle ${\alpha}_{r}$ relative to the theoretical angle ${\alpha}_{0}$, so Eq. (2) can also be written as follow:
where ${\alpha}_{0}$ is a constant value and ${\alpha}_{r}$ is a uniform random variable in a small limited range that decide by the clearance of each joint.
So, the clearances caused deployment error in the free state can be obtained:
where ${\left[{x}_{i},{y}_{i},{z}_{i}\right]}^{T}$ is the theoretical coordinate value in fixed reference frame $o$$xyz$ of point $i$.
3.3. Deployment error model with considering forces influence
As shown in Fig. 4, the “virtual massless link” can both bear the tension and stress. It provides an extra support for the basic frame which is nonexistent in the actual stressing state of the joint with clearance and may fall into the extra singularities. Therefore, “a virtual massless rope” which is inelastic and can only bear tension is appropriate instead of “virtual massless link” to describe the clearance of joint. The basic frame is a multipledegreeoffreedom system while in the forces constrained state, the azimuth angles of “virtual massless rope” and links can be expressed as follows:
where ${\mathbf{F}}_{i}$, $i=$ 1, 2,…, $n$ denotes the force vector that acting on basic frame.
Take the infinite point along the ${\mathbf{F}}_{i}$ direction as the zero potential energy point, so the location of minimum potential energy can be achieved at the extreme position of the motion rang along the ${\mathbf{F}}_{i}$ direction:
where ${\alpha}_{i}$ is a variable of ${\theta}_{i}$ that denotes the angle between ${\mathbf{m}}_{i}$ and ${\mathbf{F}}_{i}$, ${l}_{O{A}_{i}}$ and ${l}_{{O}_{i}{B}_{i}}$ denote the lengths of the tethers at both ends of link ${A}_{i}{B}_{i}$, respectively.
Fig. 7Link model with rope tied and force constrained
For the microgravity conditions and the slowly deploy speed of satellite antenna, the gravity and inertial forces can be ignored, so the basic module can be approximated as a system only bearing spring force and cabletension. The cabletension can be equivalent to the forces that point to the centre of the basic module, as shown in Fig. 8.
Fig. 8The model of basic frame with joint clearance and forces constrained
The spring force ${\mathbf{F}}_{1}$ acts on slider A along the positive direction of $z$ axis. The resultant force of cabletension ${\mathbf{F}}_{23}$ directly acts on link FG along the negative direction of $x$ axis. So, the basic fame with these forces constrained should not only meet the geometrical relationships as described in the free state, but also need to meet the following equilibrium criterion:
where ${m}_{1}$ and ${m}_{2}$ are the position vectors in fixed reference frame $o$$xyz$ of the middle points of revolute joint $B$ and link ${F}^{\text{'}}{G}^{\text{'}}$, respectively. The weights ${k}_{T}$ and ${k}_{z}$ are for the spring force and cabletension, where ${k}_{T}+{k}_{z}=$ 1. By choosing ${k}_{T}=$ 1 an equilibrium criterion of the system only with spring force constrained is found, while setting ${k}_{z}=$ 1 enables one to obtain an equilibrium criterion of the system only with cabletension constrained. The values of the weights ${k}_{T}$ and ${k}_{z}$ can be chosen in order to obtain the equilibrium criterion of the system with various spring forces and cabletensions constrained.
The deployment error with considering forces influence ${\delta}_{c}$ can be obtained from Eq. (19) by subjecting to the equilibrium criterion as described in Eq. (22).
4. Deployment error solutions
4.1. geometric parameters
According to the selected fit tolerances, all joint clearances ${r}_{i}$ are within the range of 0.010.04 mm. Table 1 gives the theory coordinates of joints in the fixed reference frame $o$$xyz$.
Table 1Theory coordinates of joints
Joint  Theory coordinate (mm) 
P  (30.00, 0, 500.00) 
Q  (30.00, 0, 480.00) 
R  (85.00, 0, 0) 
B  (30.00, 0, 231.59) 
C  (130.91, 0, 454.85) 
D  (309.67, 0, 410.29) 
E  (1496.36, 0, 114.49) 
F  (1516.30, 0, 116.11) 
G  (1418.53, 0, 563.89) 
H  (324.23, 0, 406.66) 
M2  (1489.00, 0, 721.24) 
4.2. Solution of the deployment error in free state
Using the MonteCarlo method to simulate the values of joint clearances and azimuth angles, i.e. it provides various kinds (the number is 104 in this study) of the deployment configurations of basic frame. Then calculate the deployment errors ${\delta}_{f}$ of the obtained configurations. Fig. 9 shows the results. The mean errors in the $x$ axis and $z$ axis direction are ${\mu}_{fx}=$ –0.0056 mm and ${\mu}_{fz}=$ 0.0242 mm, respectively. And the standard deviations are ${\sigma}_{fx}=$ 0.0344 mm and ${\sigma}_{fz}=$ 0.0915 mm, the maximum errors are ${\delta}_{fx}=$ 0.1111 mm and ${\delta}_{fz}=$ 0.3393 mm. The mean, standard deviation and maximum value of composition errors (the distance between actual location and theoretical location) of the fixed position of support cables M2 are ${\mu}_{fl}=$ 0.0833 mm, ${\sigma}_{fl}=$ 0.0568 mm and ${\delta}_{fl}=$ 0.3537 mm, respectively. This shows that, the joint clearances have a great effect on the deployment error in the $z$ axis direction.
Fig. 9Deployment errors distribution in the free state
a) Deployment errors distribution in the $x$ axis direction
b) Deployment errors distribution in the $z$ axis direction
c) Composition deployment errors distribution
4.3. Solution of the deployment error with considering forces influence
The deployment configuration of basic frame with forces constrained should subject the constraint as defined in Eq. 22. Therefore, the solution procedure of error ${\delta}_{c}$ involves an optimization problem of the nonlinear objective function with multiple constraints and strong coupling. So, the MonteCarlo method and genetic algorithm are combined (as shown in Fig. 10) for solving the deployment error ${\delta}_{c}$.
Constructing the fitness function by objective function $f\left({f}_{{\theta}_{i}}\right)$, as shown in Eq. (23) and Eq. (24):
where $c$ is the conservative value of the maximum value of $f\left({f}_{{\theta}_{i}}\right)$ that can be obtained from formula $c={k}_{T}{z}_{Q}{k}_{z}{x}_{G}$, k is the control parameter of fitness, and ${f}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\text{'}}$, ${f}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\text{'}}$, ${f}_{avg}^{\text{'}}$ denote respectively the maximum, minimum and average values of fitness, which are solutions from Eq. (23) within contemporary populations. When ${f}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\text{'}}{f}_{avg}^{\text{'}}<{f}_{avg}^{\text{'}}{f}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\text{'}}$, indicating that the fewer optimal individuals, so should make the optimal one has the distance of maximum to $k+$ 1 times from the worst one by Eq. (24). And as a result, to enhance the ability of superior individuals.
In the solution procedure, set the population size to 50, the maximum number of generation to 100, and the simulation number of MonteCarlo method to 104. The mean ${\mu}_{c}$ and the standard deviation ${\sigma}_{c}$ of the obtained deployment errors obey the variation laws versus ${k}_{T}$, as shown in Fig. 11. The standard deviations ${\sigma}_{cx}$ and ${\sigma}_{cz}$ respectively of the deployment errors in the $x$ axis and $z$ axis direction are monotone increasing with the increasing of ${k}_{T}$. The standard deviation ${\sigma}_{cl}$ of composition errors is monotone increasing while ${k}_{T}\le $ 0.5, and it tend to be stable with a slightly reduction as ${k}_{T}>$ 0.5. While ${k}_{T}=$ 1,${\sigma}_{cx}=$ 0.0302 mm, ${\sigma}_{cz}=$ 0.0788 mm, ${\sigma}_{cl}=$ 0.0489 mm, which are slightly less than the standard deviations of the errors in free state. It suggests that the spring force operating a micro effect on reducing the range of deployment errors, i.e. it has a slightly effect on repeat deployment accuracy.
Fig. 10Solution procedure of deployment error with considering forces influence
Fig. 11The variation laws of error characteristics
a) The variation laws of standard deviations
b) The variation laws of mean errors
Also, it can be found that the deployment error fluctuates in a smaller scope with the increasing of cabletension, especially when ${k}_{T}<$ 0.2, the standard deviation is gradually stabilising. So, it is effectively in improving the repeat accuracy by increasing cabletension. Actually, the designed cabletension $\left{\mathbf{F}}_{23}\right$ and spring force $\left{\mathbf{F}}_{1}\right$ optimal to have the difference of at least four times, and the spring force should not be enhanced while other requirements of basic frame are satisfied for decreasing the required cabletension. The minimum standard deviations with the values of ${\sigma}_{cx}=$ 0.0227 mm, ${\sigma}_{cz}=$ 0.0687 mm and ${\sigma}_{cl}=$ 0.0427 mm are obtained while ${k}_{T}=$ 0, which are decreased respectively by 34 %, 25 % and 25 % as comparing with the values in free state. The corresponding mean errors are ${\mu}_{cx}=$ –0.0321 mm (5.7 times than the free state), ${\mu}_{cz}=$ 0.0261 mm, ${\mu}_{cl}=$ 0.0716 mm, respectively. Fig. 12 shows the probability density distribution, obviously, the errors are more centralized, and the mean error in the $x$ axis direction is much larger compared with the free state (as shown in Fig. 9). It indicates that the cabletension offsets the links along the force direction. In addition, the actual deployment accuracy can be improved by optimizing the deployment position of point M2 to the new M2 (1489.0321, 0, 721.2139), which has the offsets of ${\mu}_{cx}$ and ${\mu}_{cz}$.
Fig. 12Deployment errors distribution with only considering cabletension influence
a) Deployment errors distribution in the $x$ axis direction
b) Deployment errors distribution in the $z$ axis direction
c) Composition deployment errors distribution
5. Conclusions
In this study, the “virtual massless links” are used to replace the joint clearances of deployable mechanism which is in the free state. Based on the same constraints, the deployable mechanism with multiloop structure is decomposed into some singleloop structures, and the relationship of them is obtained by DH theory. Furthermore, the deployment error distribution regularities are revealed benefit from the applying of MonteCarlo method. Besides, the forces effect on deployment error is investigated. First, the equivalent model that without any extra singular configuration of the deployable mechanism with joint clearances and forces constrained is obtained by using “virtual massless rope” instead of “virtual massless link”. Then, the stable configuration criterion of mechanism with forces constrained is presented. Finally, the deployment error with forces influence is solved by combining the MonteCarlo method and genetic algorithm. And a comparative analysis between this result and the deployment error in free state is provided. It indicates that the acting forces are effective in improving the repetitive deployment accuracy of mechanism with joint clearance. In addition, an error correction approach is introduced for further improving the deployment accuracy.
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About this article
The study of the paper is supported by National Natural Science Foundation of P. R. China (No. 51675264) and Basic Scientific Research Business Expenses of Nanjing University of Aeronautics and Astronautics (No. NE2017105).
Lin Fei presents the method described in this paper, and is responsible for writing. Chen Meng provides guidance for this paper. Chen Chuanzhi is responsible for programming. Chen Jinbao provides the research background of this paper. Dong Zirui is responsible for literature review. Zhou Jinhua is responsible for data processing.