Optimum kinematic – dynamic performance of the reconfigurable delta robot through genetic algorithm optimization

2.1. Direct kinematic

The input to the problem is the position of the end effector in direct kinematics (DK), and the output is the position of the moving platform. Direct kinematics is commonly used in robotics for simulation and laboratory tasks. A schematic diagram of the direct kinematics solution is provided in the Fig. 2 designed the delta robot, it is essential to understand specific values and dimensions, which, when combined with the fewest number of parameters, can fully define the robot’s structure and workspace. In delta robots, there are typically four design parameters. The Fig. 1 clearly illustrates these four design parameters in Table 1.

Given our goal of minimizing the number of parameters in the design, the Fig. 1 demonstrates that the radii of the base and moving circles are aligned in the same direction, indicating that the moving circle does not rotate relative to the base circle. Therefore, to simplify the parameters are considered the difference between the values of these two radii in our calculations. In reality, when solving for the inverse kinematics and the working space, only the difference between these two parameters is relevant. As a result, that will have three key design parameters, as shown in the Fig. 2 establishes a vector relationship for the robot's parallelogram link, the Eq. (8) represents this relationship, which effectively describes the length of the parallelogram arm in the Fig. 2:

To substitute the coordinates of the points $x_0$, $y_0$, and $z_0$ from Fig. 4 into Eq. (1). By assigning the values 1, 2, and 3 to the $i$th parameter, an independent relationship is derived:

For the three branches of the desired robot, the corresponding relationships are established by substituting values 1, 2, and 3 for the $i$th parameter in the Eq. (10):

Fig. 2Schematic half-side of Delta robot

2.2. Inverse kinematic

Inverse kinematics is a critical step in robotic analysis, as the selected dimensions and angles directly influence the mechanism’s structure and determine the potential for singularities. In this process, the objective is to calculate the operator’s position in space by solving a set of Eq. (10). A schematic representation of the inverse kinematics solution process is depicted in Fig. 1. The length of the line BC in space is determined by the difference between the coordinates of points C and B, which can be calculated using the spatial distance formula:

where $x$, $y$, and $z$ are the coordinates of the end-effector at point $P$, and $x_0$, $y_0$, and $z_0$ represent the coordinates of point $B$. This relationship can be formulated for each of the kinematic chains as follows. The coordinates $x_0$, $y_0$, and $z_0$ differ for each chain, while the value of $r$ remains constant across all three chains, representing the length of the line BC. Substituting this relationship into Eq. (12) yields the solution for $i=$1, 2, 3:

To solve the inverse kinematics problem, the goal is to determine the angles of the actuators based on the known $x$, $y$, and $z$ coordinates of the end-effector. To accomplish this, that utilizes a method involving variable substitution and polynomial equations. Specifically, by substituting the corresponding values for the first kinematic chain to derive its actuator angle. This process will be repeated for the remaining chains to fully resolve the inverse kinematics in Eq. (13):

After determining the coefficients of Eq. (10), these coefficients are modified as follows to simplify the calculations. As a result, the Eq. (14) are reduced to a quadratic form, which can then be solved straightforwardly using standard methods such as the quadratic formula:

In the same manner, the angles for all three operators can be determined. After solving the inverse kinematics problem and calculating the operator values, each operator will have two possible solutions, resulting in a total of eight potential solutions for the inverse kinematics problem when considering the three operators. However, only one solution is physically feasible and acceptable. This is because all of the robot’s branches must be oriented outward, and any inward orientation of the robot arms would cause them to pass through a singular point, which is not permissible. As discussed in the inverse kinematics model, the primary objective is to derive joint variables or active interface angles from the Cartesian space variables, which represent the robot’s final position. Due to the inherent nature of parallel robots, inverse kinematics models typically yield multiple solutions. Consequently, the mathematical model derived here also provides eight possible solutions. However, through systematic analysis of these solutions, the correct one can be identified. In the program written to solve the inverse kinematics, specific conditions are implemented to ensure that the correct solution is selected in each case. This can be achieved by carefully examining Eq. (15), as it forms the basis for determining the valid solution:

Now, the aforementioned relationship is rearranged in terms of the joint space variables, yielding the following form:

where in:

Now, by employing the half-angle tangent solution method, the above trigonometric Eq. (18) can be expressed as follows:

2.3. Direct kinematic

To derive the direct kinematic relationships, which serve as the foundation for establishing the relationship between the robot’s joint space and its Cartesian coordinates. This Eq. (19) guides in determining the position of the robot's moving platform based on the known joint variables:

From a physical perspective, the set of equations described above, incorporating the parameters $X_p$, $Y_p$, and $Z_p$, represents the geometric intersection of three spheres. Typically, this configuration results in two distinct solutions. Specifically, the intersection of two spheres forms a circle, and the intersection of this circle with a third sphere yields two points, which correspond to the solutions of the Eq. (1). To solve this Eq. (1), the system can be expanded and reformulated as three equations with three unknown variables: $X_p$, $Y_p$, and $Z_p$. This approach allows for determining the Cartesian coordinates of the moving platform, as demonstrated Eq. (20):

As previously mentioned, solving the direct kinematic equations results in two possible solutions. Since it has been taken the coordinate origin as the center of the fixed plane and treat the robot as if suspended from above, all robot positions within the workspace will have negative Z-coordinates. Therefore, it is sufficient to choose the solution that leads to a negative $Z_p$ value from these two results. Once $Z_p$ is determined, the corresponding $X_p$ and $Y_p$ values can be uniquely obtained. As discussed in inverse kinematics, the solution is considered in four possible scenarios:

1) Two points are obtained from the intersection of three spheres, resulting in two independent solution branches.

2) One of the spheres is tangent to the circle formed by the intersection of the other two spheres, yielding a single solution.

3) The centers of two spheres coincide, resulting in an infinite number of solutions.

4) The three spheres do not intersect, indicating that no solution exists.

After presenting the position kinematics of the Delta robot is now paid attention to the velocity kinematics. In general, analyzing a robot's velocity kinematics leads to the calculation of the Jacobian matrix. This matrix is essential in the field of robotics, as it is used not only in kinematic calculations but also provides valuable information about the robot's motion. The Jacobian matrix is defined as follows:

The final end-effector position vector is represented by $x$, while the vector of joint variables is represented by q. This defines the Jacobian matrix as the inverse of the standard Jacobian matrix:

where in:

This notation is simply a convention, indicating that once the $J_x$ and $J_q$ are obtained as matrices, the core work is complete. From this point onward, the standard form of the Jacobian matrix can be derived.

2.4. Calculation of ${\mathit{J}}_{\mathit{x}}$ and ${\mathit{J}}_{\mathit{y}}$ matrices

To calculate the Jacobian matrix again, refer to the Fig. 2 is expressed the closed-loop equation for the $i$th base as follows:

After differentiating this Eq. (25) with respect to time, it obtains:

That $V_p$ represents the linear velocity of the moving plate:

The ${\omega }_{ij}$ represents the angular velocity of the $j$th link in chain $i$. For this robot, the input vector is $\dot{q}={\left[\begin{array}{ccc}{\dot{\theta }}_{11}& {\dot{\theta }}_{12}& {\dot{\theta }}_{13}\end{array}\right]}^T$, and the output vector is$V_p={\left[\begin{array}{ccc}{v}_{p,x}& v_{p,y}& v_{p,z}\end{array}\right]}^T$. Therefore, in Eq. (26) corresponds the ${\omega }_{ij}$ to the input joint velocities, and the ${\omega }_{2i}$ are associated with the passive joints and should be excluded from the relation. To eliminate the velocities of the passive joints, and both sides of the Eq. (25) that are multiplied by the $b_i$ vector:

By expressing Eq. (27) in the coordinate system $\left(x_i{y}_i{z}_i\right):$

By substituting the above values in Eq. (27) obtains the simplified relationships:

where in:

By writing Eq. (29) for $i=$ 1, 2, 3, which leads to three scalar equations, and arranging them in a matrix form obtains the desired result in Eq. (30):

where in:

2.5. Singularity

Singularity is a critical issue in robot design, as it helps identify areas where the robot’s movement may encounter difficulties. To analyze singularity, the first step is to calculate the Jacobian matrix. The Jacobian matrix typically defines the relationship between Cartesian space and joint space. Referring to Fig. 2, that writes the vector relation for a closed loop or one of the branches:

By taking the time derivative of Eq. (33) obtains the following Eq. (33):

Now, to eliminate the angular velocity of the second arm $w_2$ internalize the entire relationship in $L_b$:

After simplifying the mathematics, it obtains:

The Jacobian matrix derived from the Eq. (35) is critical because the robot experiences singularities at the points where the determinant of the Jacobian matrix equals zero. These singular points represent configurations where the robot loses degrees of freedom and cannot move freely. It is essential to avoid operating the robot at these points. By analyzing the Jacobian matrix, to identify the singular points as follows.

2.6. Delta robot’s dynamic

To solve the dynamic problem begins by making certain assumptions, followed by solving Lagrange's equation (Eq. (37)). For the solution is assumed that the mass of the robot’s second link, or the parallelogram mechanism, behaves as a point mass, and its moment of inertia is negligible compared to other components. In reality, the second link of the robot is significantly lighter than the first. Table 3 presents the weight specifications of two delta robots.

The second link’s mass is around 40 to 60 grams, which will not have a significant effect in practice. Now, using this assumption is considered the second link as a point and calculate the entire robots kinetic energy and potential. The total number of variables in this problem is 6, of which 3 are dependent variables:

In this case, Lagrange’s equation becomes:

Calculate the kinetic and potential energy using the following formulas:

where $m_b$, $m_{fb}$, and $m_{nt}$ represent the mass of the first arm, follower arm, moving plate, respectively. To obtain the Lagrange coefficients must be solved the constraint equation, which is shown below:

The first set of Lagrange equations (in Eq. (40)), including the Lagrange coefficients, can be expressed as follows:

In the above relations, the only unknowns are the Lagrange coefficients, which can be determined by solving the system of three equations with three unknowns. The second set of Lagrange equations involves the operators and the corresponding required forces. By solving the Lagrange equations for $q_4$ to $q_6$, the required torque is determined as a function of the operator’s angles:

2.7. Reconfigurable delta robot

Figs. 1 and 2 focus on which represent a robot with geometric and dynamic characteristics. First, the ${\alpha }_4$ is calculated:

The parameters in Eq. (45) can be expressed as follow.

Now, to simplify Eq. (45) and arrive at Eq. (46):

The constants in Eq. (46) can be expressed separately as follows:

The fixed coordinate system $X_0{Y}_0{Z}_0$ is considered the reference system. The coordinates of the point force $F_i=\left(F_{xi},F_{yi},F_{zi}\right)$, corresponding to the point $A_i=\left(A_{xi},A_{yi},A_{zi}\right)$, are dependent on the angular displacement, which is related to the engine. The angle of the square frame plays a crucial role. Additionally, the coordinates of $B_i=\left(B_{xi},B_{yi},B_{zi}\right)$, as shown in the Fig. 2, are generally calculated using the following relationships [34]:

The angle ${\theta }_i$ is the angular coordinate of each kinematic chain $R$:

The distance between the fixed coordinates $X_0{Y}_0{Z}_0$ and point $F_i$ is represented by $R_f$. Based on the previously established relationship, $B_{y_i}$ is calculated as follows:

A moving triangular platform is considered at points $C_1$, $C_2$, and $C_3$ with distance $r$ and center $p$, where the coordinates of $C_1=\left(X,Y,Z\right)$ are given, and the coordinates of $C_2$ and $C_3$can be calculated as follows:

The angle represents the kinematic chain of points $C_2$ and $C_3$ and based on the calculation of $X$, $Y$, $Z$ coordinates, three equations resulting from the following relationship can be derived [34]:

Hence:

The $K_{i4}$ can also be obtained for $j=$1, 2, 3, and 4 using the general and global parameters and coordinates of the mechanism. For $X$, $Y$, and $Z$, Eq. (51) can be described as follows: