Deployment accuracy analysis of cable-strut deployable mechanism with joint clearances and forces constrained

3.1. Equivalent model of joint clearance

The basic frame pose of ETS-Ⅷ 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 multi-rigid-body 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 normal-distributed 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 $\max \left(r_i\right)$ and $\min \left(r_i\right)$ 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 D-H 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 cable-tension, 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 D-H 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 single-loop 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 single-loop

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 D-H 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 multi-closed 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 non-existent 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 multiple-degree-of-freedom 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 cable-tension. The cable-tension 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 cable-tension ${\mathbf{F}}_{23}$ 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 cable-tension, 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 cable-tension 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 cable-tensions 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).

4.1. geometric parameters

According to the selected fit tolerances, all joint clearances $r_i$ are within the range of 0.01-0.04 mm. Table 1 gives the theory coordinates of joints in the fixed reference frame $o$-$xyz$.

4.2. Solution of the deployment error in free state

Using the Monte-Carlo 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 Monte-Carlo 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 Monte-Carlo 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 cable-tension, especially when $k_T<$ 0.2, the standard deviation is gradually stabilising. So, it is effectively in improving the repeat accuracy by increasing cable-tension. Actually, the designed cable-tension $\left|{\mathbf{F}}_{23}\right|$ 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 cable-tension. 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 cable-tension 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 cable-tension influence

a) Deployment errors distribution in the $x$ axis direction

b) Deployment errors distribution in the $z$ axis direction

c) Composition deployment errors distribution