8
Neural Network Based Learning and Control Co‐Design for Stewart Platform Control
8.1 Introduction
Kinematically redundant manipulators [39] are those manipulators that prove to have sufficiently higher degrees of freedom (DOFs) than required for positioning and for orientation of the platform. With the advances in the field of robotic technologies, robotic manipulators are widely used in the applications of factory automation which are required to carry out continuous and delayed work, such as lifting and transporting radioactive substances and executing the work in hazardous, scattered or packed environments. In comparison with nonredundant manipulators, redundant ones offer extra DOFs [75], and are often used to improve the dexterity, in order to work efficiently by avoiding collisions with obstacles. Research in the field of kinematically redundant manipulators has gained popularity due to their ability to avoid obstacles, their internal singular configurations, and their ability to optimize the performance of the workspace and the end‐effector motion task. Among various types of redundant manipulators, parallel ones, which usually feature higher rigidity, higher precision and higher response speed than serial ones, have received popular applications in flight simulators, electrostatic magnetic lenses, etc. However, how to efficiently control the motion of redundant parallel manipulators, especially in the situation with parameter uncertainties, or even unknown, is of great practical significance and remains a challenging research problem. Parallel redundant manipulators are broadly classified as parallel manipulators where the task space coordinates are lower than the number of actuators. These manipulators are found in many industrial applications such as robotics arms, surgical robots, and so on. These manipulators offer greater advantages when incorporated because they make the structure flexible, faster, and lighter thereby improving the Cartesian stiffness and optimizing the distribution of the force. Due to the advantages of high speed and high acceleration, parallel manipulators have been studied and implemented widely.
Redundant manipulators follow similar dynamic equations as for the serial ones. Therefore, extending design studies of serial manipulators to parallel manipulators is fairly obvious. However, designing the dynamics of the parallel manipulator is more complicated. Moreover, it is quite challenging to identify the unknown parameter which describes the dynamic behavior or properties of the system. Hence, the control schemes of the existing traditional methods for serial manipulators cannot be extended in real‐time control applications to parallel redundant manipulators.
Parallel manipulators are confined to age‐old and basic problems of identification and classification of singularities. A lot of work is developed using mathematical tools borrowed from serial manipulators for local analysis and to resolve the problem of singularities. Gosselin was the first to define, study, analyze and report the singularity problem for the closed‐loop kinematic chains [97]. The structure and behavior of the singularity problem for the parallel manipulator is indeed complex and challenging. Many studies are incorporated to address the kinematic manipulability measure for design and control of parallel mechanisms. Recurrent neural networks, as a powerful parallel computation method, are proven effective and efficient for the applications of real‐time solutions to the inverse kinematics problem. In the literature of the past decade, a variety of dynamical system solvers have been proposed to resolve the problems of online constrained quadratic programming, including the primal dual network, Lagrange neural networks, the gradient network, and the projected network. There are also traditional approaches which consider joint and velocity constraints. For expressing a general solution in the form of redundant join velocities, the Gram–Schmidt orthogonalization procedure is utilized.
Constraints in soft computing techniques introduce two main categories of difficulties in obtaining the solution to the problem. First, the challenge is of the independence where the coordinates are not independent; and secondly, a priori information of the constraints forces is not sufficiently provided and they are regarded among the unknowns of the system. Hence, control of the Stewart platform as a constraint system becomes complicated due to the complex nature of the neural dynamics. Another limiting factor in conventional robotic manipulation research using dual neural network approaches is the requirement for a design model, which involves constructing a mathematical model that highlights the controlled dynamics of the model. The initial stage of the design usually requires the interd dependence between different parts and their historical dependence on previous states to be established. A later stage involves the design of the analytical controller with the mathematically modeled system dynamics. Although the designed control law gives a promising performance for the mathematical model, this might not be the case in real‐time applications as the exact representation of the model is hard to obtain. This may be due to various reasons, e.g. the sheer complexity of the designed model or the uncertainty involved in the area. However, modeling of the feedback control for the physical system brings about the tradeoff between the ease of modeling and the precision of the physical system in matching. Due to the improvement of the parameters in the controller depending upon the convergence and stability factors, adaptive techniques usually demonstrate outstanding results in the face of complex systems. This motivates us to devise an adaptive and model‐free neural controller to steer the motion of a Stewart platform.
In this chapter we aim to close the gap between the two research areas of model‐free recurrent neural networks for machine learning and model‐based dual neural networks for accurate model control. The proposed network interacts with the learning and control parts. An excitation noise is added to avoid the learning degradation. This deliberate design offers precise convergence of the estimated variables to their true values. The stability of the network is proved theoretically and by simulations and it is proved that the bounded error is found to be arbitrarily small by scaling the additive noise.
8.2 Kinematic Modeling of Stewart Platforms
A Stewart platform, as sketched in Figure 8.1, consists of a 6‐DOF platform comprising two plates, namely a fixed base plate and a flexible or moving top plate, which is in turn connected to a series of prismatic actuators and passive joints. Each of the prismatic actuators is connected by a spherical joint to the base plate. A base plate is connected by universal joints to each of its actuators. This specific arrangement of actuators and joints allows the top moving plate to move on either side depending upon the lengths of the prismatic actuators or leg joints.
Figure 8.1 Schematic of a Stewart platform.
8.2.1 Geometric Relation
There are two coordinate systems associated with a Stewart platform, namely a base coordinate system which is fixed as a global system and a moving coordinate system of the platform. We use 
to distinguish a variable defined in the base coordinate system from the corresponding one defined as 
in the global coordinate. In Figure 8.2, the position vectors 
indicate the position of the center of the universal joint of the leg. Thus, 
, defined as the position vectors, represent the moving platform positions in global coordinates of the base's ith connection point. The vector 
represents the ith leg of the actuator pointing from the base to the platform. The global coordinates and the platform coordinates are fixed to the base and the mobile platform, respectively. 
for 
dictates the position in global coordinates of the ith connection point on the base. 
for 
denotes the position in platform coordinates of the ith connection point on the platform. Hence 
are defined to represent the global coordinate (Figure 8.2). 
for 
is defined to represent the ith leg vector from the base to the platform. For 
in platform coordinates, the corresponding global coordinates 
can be derived after a translational and rotational transformation:
where 
represents the global coordinates of the origin position in the platform coordinate, and corresponding to the translational transformation, 
is the rotational matrix, which is predominantly defined by the Euler angles 
,
Following Equation (8.1), as to the ith connection point on the platform, i.e. the ones with 
in the global coordinates or the ones with 
in the platform coordinates, we have
Therefore, the ith leg vector can be further expressed as
For the vector 
, we define 
to represent its length. Accordingly, we have
It is to be observed that both 
and 
are constants. Henceforth, they can be derived by the geometric structure. 
denotes the coordinate frame which defines the translation of the platform, and 
represents a function of the Euler angles 
, denoting the rotation of the Stewart platform. The right‐hand side of (8.5) depends on the pose variables of the Stewart platform 
while the left‐hand side is the length of the leg, which is in turn controlled for actuation. Hence we highlight that in this way, Equation (8.5) for 
defines the relationship of kinematics between the actuation variables and the pose variables. Now, for a defined six‐dimensional reference position, the desired leg length 
can be directly obtained from (8.5). However, in real‐time applications, the reference positions are usually not defined as six‐dimensional. This could be explained by an example from the medical industry: in surgical applications which include a Stewart platform, doctors only care about the position of an end‐effector on the platform, and are not concerned about its orientation. Owing to this fact, usually in these scenarios the reference is three‐dimensional and therefore we have three additional DOFs as redundancy. For these kind of challenges, we ultimately have an infinite number of feasible solutions of 
for 
to reach the reference. Among all of the feasible options and solutions, we may be able to identify a unique solution, which outperforms others and existing solutions in terms of certain criteria of optimization. This intuitive and mathematical approach analyzes and motivates the modeling of the optimization problem and identifies the optimal solution for the improvised performance. Due to presence of the nonlinearity of Equation (8.5), treating Equation (8.5) directly is technically impractical and prohibitive. Hence, we formulate the problem in terms of velocity space to explore the linearity approximation rather than studying the problem in position space for a direct solution.
Figure 8.2 Stewart platform geometric representation. The gray triangle at the top and the gray hexagon at the bottom represent the moving top plate and the fixed base plate, respectively.
8.2.2 Velocity Space Resolution
For simplicity, Equation (8.5) is rearranged as follows:
We define, represent and obtain the velocity space relations by computing the time derivative on both sides of (8.6), which yields
Recall that both 
and 
are constants and their time derivatives, 
and 
, are equivalent to zero. For the rotational matrix 
, we consider the the above discussed preliminary equations, and the following represents the time derivative property
In this way, the moving platform rotation matrix coordinate system with respect to base platform is achieved. However, the position vector specified at the origin of the moving platform denotes the translation vector with respect to the base platform.
Substituting (8.9) into (8.7) yields
As mentioned, in the above equation, Equation (8.4) is used for the derivation in the second and the penultimate lines, respectively. Noticing that 
could be guaranteed by the mechanical structure, we have the following result
For the six‐dimensional vector 
, we have the compact matrix form as follows
where
Equation (8.12) projects the kinematic relation of a Stewart platform with 6 DOFS from the velocity of the pose variables to the speed of the legs.
8.3 Recurrent Neural Network Design
8.3.1 Problem Formulation from an Optimization Perspective
In this section, we introduce a numerical and nonlinear gradient decent optimization method to resolve the real kinematic parameters from the measurement data. The digital indicators are defined as measurement devices in order to rectify and verify the location of the end‐effector of the Stewart platform. This in turn can determine the error between the desired and actual locations. The equation corresponding to kinematics of a parallel Stewart platform can be expressed as follows:
where 
represents the ith position vector in the mobile platform with respect to 
and 
represents the ith position vector with respect to the base 
. 
represents the ith link length. This equation denotes the peculiar inverse kinematic equation of a Stewart platform. In general, it is infeasible to derive a forward kinematic model for the parallel manipulators due to nonavailability of a closed form solution, and some numeral algorithms must be incorporated to derive the parameters of the forward kinematics.
The nonlinear or nonconvex relationship between 
and 
to the position variable is affine to 
. We observed in our analysis that 
can be solved directly from the compact matrix form equation (8.12). The reference velocity constrained in reduced dimensions for the equality model is defined as
where 
is the reference vector with 
. The pre‐given transformation matrix is 
. The optimized solution is defined as follows:
where the symmetric matrices 
and 
are both constant and positive definite, and the speed of the platform legs to be control is denoted as 
. In practice, the term 
, which is in a quadratic form, specifies the kinematic energy when choosing 
properly, and the input power can be characterized by 
. The formulation of the objective function is consistent with the convention of control theory in defining quadratic cost functions. The actuator can directly change the value of the decision variable 
. Its value is under physical constraints, which are modeled as inequalities in the following form,
In this expression, 
and 
with 
as an integer. It is noteworthy that it is not imposed on the variable 
for the constraint as in the planning stage it usually has already been specified. In summary, with Equation (8.16) as the object function, and (8.17) as physical constraints, and also with the nonlinear dynamic equation constraints, a constrained programming can thus be formulated to solve the control:
Since there are two types of constraints present, namely equality and inequality constraints in (8.18), it is not feasible to solve the optimizaton analytically. Incorporating traditional approaches incurs an extra penalty in terms formed by the constraints to the objective function. Resolving the problem numerically using the approach of gradient descent along the new objective function is also expensive. Hence we conclude that penalty‐based approaches are expensive and can only reach an approximate solution of the problem and therefore are not feasible to tackle error‐sensitive applications. Hence, it is worth attempting to devise a dynamic differential equation for the type of neural network to approach the solution iteratively.
8.3.2 Neural Network Dynamics
In this section, we present the neural network model used in this chapter. This is a dynamic neural model that can be described by an ordinary differential equation. The dynamics is as follows:
where 
is a scaling factor. By replacing 
with 
as obtained in (8.1), the whole system obtained so far can be expressed as
Consider a special case when the initial value of 
and 
at time 
=0 are both set at zero. In this situation, the immediate derivative of state variables can be obtained as 
. Then
Subtracting (8.20) from (8.21) yields
Since, 
,
The state variables of the neural network are shown in Figure 8.3. This figure depicts the dynamic redundancy of the neural network.
Figure 8.3 Architecture of the proposed neural network.
8.3.3 Stability
Stability is the most important issue in the dynamic systems. Nonstable systems may oscillate or even diverge. In this section we discuss the stability of the proposed architecture in learning convergence.
Define 
and use 
, where 
denotes the Frobenius norm of a matrix, as an estimation error metric. The time derivative of 
along the system dynamic is 
.
Note that the equality trace 
for any 
and 
of appropriate size is utilized in the above derivation. Thus, 
. For 
. Note that 
and is monotonically decreasing according to the above equations.
We have
We know that 
, we can multiply 
, 
and computing the expected value yields
We note that since 
. These two equations lead to the following result: 
, which further implies 
.
8.3.4 Optimality
This section shows the optimal solution of the original optimization problem can be arrived at by converging to the equilibrium point of the dynamic neural network (8.19).
8.4 Numerical Investigation
To demonstrate the efficiency and effectiveness of the proposed model‐free neural network approach applied to redundancy resolution of the Stewart platform, we implemented it in MATLAB.
8.4.1 Setups
A Stewart platform with the following specifications will be considered: leg connectors are located around a circle with radius of 1.0 m at 
, 
, 
, 
, 
, and 
in the platform coordinate, and the leg connectors are located around the circle with radius of 0.75 m at 
, 
, 
, 
, 
, and 
on the fixed base. For the ease of simulations the end‐effector is rotated and placed at the origin with respect to the platform coordinate. The total expected redundancy is assumed to be 3, for the position tracking in the three‐dimensional space. The input and output dimensions are 6 and 3, respectively.
The desired angular motion speed is set as 0.2 rad/s. The control scaling factor 
of the neural model is set as 
and the learning scaling factor 
is set as 
. The excitation signal 
is set as random noise with zero mean and deviation of 
. The basic idea is to set the noise at small value to ensure a minimal impact on the system performance.
In the simulation, we consider two tracking trajectories, namely, a square path and a circular path.
8.4.2 Circular Trajectory
In this section we simulate the tracking of a smooth circular path using the model‐free approach. It is desired to follow the path of a circle at the minimal speed of 2 m/s. The circle is centered at 
with a radius of 0.08 m, and has a revolutionary angle around the 
axis for 
. In the simulation setup, 
and 
are chosen as the identity matrix. The value of the matrix 
is computed in real‐time accordingly. The matrix 
is chosen as 
so that the position tracking requirements are obtained. In real time, the leg actuation speed is limited to a certain range as it is associated with the real‐time constraints of the actuators.
The results are obtained by executing the simulation for 2 s. The Figure 8.4a shows the completed tracks in circular motion of an end‐effector with the least tracking error as shown in Figure 8.4b. The position tracking error components 
, and 
are plotted along the 
, and 
axes of the base frame of the platform. The errors depicted in the figure are less than 0.015 m in amplitude. This path tracking simulation demonstrated the capability of the proposed model for resolving the kinematic redundancy of the physically constrained Stewart platform. The input motion for the legs is shown in Figure 8.5 which depicts the time evolution of the Stewart platform state variables, e.g. three Euler orientations of the platform, the position of the end‐effector and the leg speed and its related length. The coordinates 
and 
of the attached moving frame start from zero and 
varies between 0.005 and 
m. Figure 8.5c states the bound 
(
) for action speed. It can be observed that 
converges within the boundary region of [
0.25, 0.25].
Figure 8.4 Tracking of a circular motion: (a) The tracking trajectory of the end‐effector and (b) the time history of the position tracking error.
Figure 8.5 The time evolution of the Stewart platform state variables in the case of circular motion tracking. (a) Orientation of the platform 
; (b) control action 
; and (c) leg length 
.
8.4.3 Square Trajectory
In this section we will discuss the simulated results of the square trajectory incorporating our model‐free approach. In circular motion the path traveled by the end‐effector in 
. However, in a square trajectory the path is nonsmooth switching from one straight line to the next, and how to reach timely control becomes a challenging issue. The end‐effector follows the trajectory which is centered at [0.15, 0.075, 0.74] with an edge length of 0.08 m, at a speed of 1.0 m/s. The revolution angle of the square around the 
axis is 
. The chosen parameters 
and 
, 
and 
have the same values as mentioned for the circular motion. The parameter 
is computed in real‐time. The limit of speed 
m/s and 
, and 
. The tracking results for the square trajectory are shown in Figures 8.6 and 8.7 (position tracking error vector [
, 
, 
] is very small).
Figure 8.6 Tracking of a square motion. (a) End‐effector trajectory and (b) position tracking error.
Figure 8.7 The time evolution of the platform state variables in the case of square motion tracking. (a) Orientation of the platform 
; (b) control action 
; and (c) leg length 
.
8.5 Summary
A model‐free dual neural network is proposed to investigate and resolve the redundancy resolution problem of manipulators. In this chapter we also establish a dynamic model for a model‐free network which is designed for a general case of a modular manipulator. The proposed controller model is designed such that a priori knowledge is not required for dynamic parameters and can suppress bounded external disturbances effectively in the presence of the external noise added. The instability problem caused by self‐motions of the redundant robot can be resolved by the presented dual‐neural control algorithm. Theoretical results are presented to verify the stability of the proposed models. The simulation is carried out on a redundant manipulator, which has verified the effectiveness of the dynamic modeling method and the controller design method.