Optimal design and verification of the 6-DOF Stewart structure used in vehicle-mounted automatic assembly

The Stewart structure with six degrees of freedoms (6-DOF), which has the advantages of big bearing capacity, strong stiffness and high accuracy, has been widely used in the indoor automatic assembly of aeronautics and astronautics. In this paper, a novel design method for the Stewart structure with limited installation space is proposed to extend the workspace by parameter optimization using genetic algorithm, and the performance is further verified via co-simulation. Firstly, the target function, which describes the workspace that satisfies both kinematic and dynamic constraints, is built with four parameters: the joints circle diameters and the angles between adjacent joints of both bottom and top platforms. Subsequently, the genetic algorithm is used to optimize the target function with the four parameters whose bounds are limited by vehicle-mounted installation space and structure interference. Finally, the geometry model is built in LMS. Motion and is studied using co-simulation method with the hydraulic system in AMEsim and control system in Simulink. With the cylinders motion accuracy within ±0.5 mm, the final position of top platform is consistent with the theoretical calculations, and the position error could be controlled within ±0.7 mm.


Introduction
The electro-mechanical equipment of complex function and high power commonly has large scale and weight, which may certainly exceed the transportation capacity of single truck.It should be divided into several pieces during manufacturing and delivery to ensure the mobility, and assembled as a whole in preset position.In order to erect the pieces into a precise and complex electro-mechanical system, the assembly mechanism should have at least 6-DOF, big bearing capacity, strong stiffness and high accuracy, and should be easy to control.Stewart structure could meet the requirements mentioned above and thus has been widely used in aeronautics and astronautics [1,2].However, these indoor applications commonly have unconstrained installation space to overcome the inherent defect of Stewart structure of inadequate workspace [3].
Figure 1  In order to extend the workspace of vehicle-mounted Stewart structure with limited installation space, heavy weight load and complex wind force, the parameter set P should be optimized.In previous literatures, the optimization targets were mainly about manipulability, condition numbers, dexterity and isotropy, which are not very practical in actual application [4,5].Meanwhile, the accurate forward kinematic and dynamic solutions are still hard to achieve, let alone the interference analysis via mathematical method [6,7].The optimal design and verification of a Stewart-type mechanism for vehicle-mounted automatic assembly are carried out in this paper.In Section 2, a novel target function described by P is proposed to define the workspace of a Stewart mechanism, and is further optimized via genetic algorithm.With the optimized parameters, the geometry models of the assembly mechanism are built in LMS.Motion, together with the hydraulic models in AMEsim and control models in Simulink, and are studied via co-simulation in Section 3 to validate the interference, workspace and accuracy.Finally, Section 4 gives the conclusions.

Derivation of target function
For vehicle-mounted electro-mechanical equipment, when the truck stops, the initial position and posture deviations between divided position and integrated position are decided by driver skill, and are certainly random within predictable boundary.Subsequently, the top platform would move to required position along planned trajectory according to the deviations.Thus, the workspace of the Stewart structure should exceed the predictable boundary.The workspace W with specific P is confirmed along the procedure shown in Figure 2. Firstly, the studied space Q, which is defined as (-0.5:0.5;-0.5:0.5;0.06:0.3)with the unit of meter, is cut into several horizontal layers with equal interval along z-axis, just like under computed tomography, and each x-y layer would be squared off with symmetrical border, here x, y, z are separately defined as the truck wheel axis, truck advancing and gravity reverse directions.

MEIE-2023
Journal of Physics: Conference Series 2591 (2023) 012010 IOP Publishing doi:10.1088/1742-6596/2591/1/0120103 Subsequently, every cross node would be tested with the kinematic and dynamic constraints, which is separately decided by cylinder stroke and hydraulic bearing force.To be exact, the i-th cylinder vector could be described as [6]: where u, v, w is separately the rotation angles around z, y, x axis, s_u and c_u are short for sin(u) and cos(u).Thus the i-th cylinders length  could be derived.For cylinders with fixed dead zone l dead , the maximum length l i-max =l i-min + l s , where min i l  is the minimum initial length and l s is the stroke with the definition of l i-min -l dead .Obviously, the nodes with l i (i=1, 2…,6)< l i-max would meet the kinematic constrains.The cylinders force is derived by where T=[10000,2500,-250000,0,0,0] is the external load matrix, including vertical weight 250000N, horizontal wind force 10000N in x-direction and 2500N in y-direction.J is the Jacobian matrix and F is the cylinder force matrix.
where T i   is the unit vector of i-th cylinder.Thus the dynamic limitation is satisfied when where A 1 and A 2 are separately the piston and rod area of cylinder, p max is the system maximum pressure, f min is negative to represent the pull force.Finally, nodes that pass the test would be sampled to build the workspace W, and the products of x and y axis absolute values within the workspace are accumulated to build the target function: where (X W , Y W , Z W ) represents the nodes in workspace W. Finally, optimal P could be found via minimizing f(P).

Optimization using genetic algorithm
Target function f(P) is a typical multiple parameter and nonlinear function, thus classical minimization method, such as linear programming and least square method, would results in local optimal solution and cannot meet the requirement.The genetic algorithm is used here to escape from the local optimal solution [8], and could be easily realized in MATLAB Optimization toolbox.After multiple iterations,  The nodes of workspace in layers from z=0.06~0.3m is shown in Figure 3. Due to the unbalance wind forces in horizontal direction, the nodes are not symmetrically distributed.

Co-simulation framework
As a high cost and long manufacture period mechanism, the control accuracy and interference of Stewart structure should be validated by simulation firstly.Moreover, co-simulation strategy is preferred for this machinery and hydraulic integrated system.The co-simulation framework is shown in Figure 4.In order to move steadily in all 6 DOFs, the assembly trajectory is planned to accelerate in the beginning and decelerate at the end stop, while the speed in the long middle stage is constant.After trajectory planning, the motion law of cylinders could be derived via inverse kinematic in MATLAB/Simulink, and is controlled with the displacement feedback sampled in LMS.Motion.The control signal calculated in Simulink is transmitted to AMEsim to drive the servo valve and generate driving force in the cylinders.Finally, the driving forces act on the cylinder joints in LMS.Motion and lift the top platform.Meanwhile, structure interference could be examined during the animation procedure in LMS.Motion.

Co-simulation results analysis
The initial position deviations in LMS.Motion are set as 0.42 m along x-direction, -0.18 m along ydirection and 0.12 m along z-direction.The total simulation period is 16 s, while the acceleration and deceleration time are both 3.6 s.The step size is 5e -4 s, and the algorithm in Simulink is ode8.The cylinder displacements in Figure 5 change along designed trajectory to decrease the start and stop shock, and have similar form but different amplitude.Figure 6 depicts the cylinder errors.Caused by the delay of the establishment for system hydraulic pressure, cylinder displacements have a big overshot and fluctuation in the beginning, and could be constrained within ±0.5 mm in stable condition.The position deviations of the top platform in Figure 7 also have an obvious fluctuation under external load at the beginning, then the deviations in all directions decrease softly and the final error could be controlled within ±0.7 mm, which proves the validity of the inverse kinematic in Section 2.Moreover, no collision is detected during the animation in LMS.Motion, which proves that the structure has no interference.

Conclusions
With the novel proposed target function optimized by genetic algorithm, the workspace of the Stewart structure could be extended with limited installation space.The optimized parameters could guide the following structure design.
The accuracy, bearing capacity and interference of the Stewart structure could be validated easily via co-simulation.With the cylinder control error within ±0.5 mm, the position accuracy of top platform is no more than ±0.7 mm along all directions, and could meet with expectation.The simulation results indicate the feasibility of automatic assembly using the vehicle-mounted Stewart structure.
depicts a classical Stewart structure, where the bottom platform is fixed on the truck and the top platform is controlled by six cylinders to lift the mobile part mounted thereon to required position with specific orientation.Every cylinder connects the bottom and top platforms via universal joints a i and b i with two rotational DOFs.In the bottom and top platforms, D and d is separately the diameters of the universal joint cycles, O A and O B are the cycle centers, α and β are the central angles between adjacent joint points.The initial height between O A and O B is fixed duo to transport circumscription, and the parameter set P=[D, d, α, β] is mainly constrained by installation space and cylinder motion intervention.

Figure 2 .
Figure 2. The design procedure to build workspace W.

Figure 3 .
Figure 3.The nodes of workspace in different layers from z=0.06~0.3m.

Figure 7 .
Figure 7. Position errors of top platform along three axes.

Table 1 .
P=[1.6,20,1.2,30]couldbeobtained.The values of target function with optimized P and the P based on experience are listed in Table1as contrast.Target values with different P.