Inverse Kinematic Algorithm for 8-DOF Redundant Manipulators Based on Weighted Least-Norm Method and Parameterization Method

 Abstract: Eight-degree-of-freedom redundant manipulators are considered more suitable for performing special tasks in unstructured environments due to their high degree of freedom and flexibility. The 8-DOF tunnel shotcrete manipulator is a kind of equipment that is applied to the concrete spraying support in the process of tunnel excavation. Since the mechanism itself is a redundant manipulator with link offset, its inverse kinematics is extremely difficult to solve. To address the problems that the conventional redundant manipulator inverse kinematics solution is difficult to solve analytically and the numerical algorithm requires several iterations with accumulated errors, a synthesis inverse kinematics algorithm based on the weighted minimum norm method and the joint angle parameterization method is proposed to overcome these defects. Firstly, a set of optimized joint angle approximation solutions is obtained based on the weighted least norm method of joint angle limit avoidance optimization function, and secondly, two joint variables are parameterized as known parameters in the optimized approximation solutions based on the joint angle parameterization method, and further, the analytical expressions of the remaining joint angles are derived by algebraic methods. Finally, the superiority of the inverse kinematic synthesis algorithm over the joint angle parameterization method in terms of joint motion is verified by comparing numerical examples of linear motion trajectories. Numerical simulation results show that the algorithm has the advantages of avoiding joint angle limits, high computational accuracy, no cumulative error, and the algorithm can meet the general real-time motion control requirements as demonstrated by the experimental analysis of algorithm time


Introduction
Tunnel shotcrete robot is a kind of construction machinery used for concrete shotcrete support of tunnel wall after blasting, which plays an irreplaceable role in tunneling engineering. The inverse kinematics of the shotcrete robot arm is one of the key problems that restrict the realization of intelligent shotcrete spraying in tunnels, and its inverse kinematics is extremely difficult to solve due to the existence of redundant degrees of freedom, complex joint structure, and link offset of the robot arm mechanism. The degrees of freedom (DOF) of the current large tunnel shotcrete manipulator are mostly 8-DOF [1], and the inverse kinematics problem of the robot arm is a prerequisite for analyzing the motion characteristics of the mechanism and realizing its trajectory planning and motion control.
The methods for solving the inverse kinematics of redundant robots are mainly divided into closed-form solution method, numerical solution method, and intelligent algorithm. The closed-form solution method mainly includes geometric and algebraic solution methods [2], in addition to two methods that combine algebraic and geometric methods, namely the arm angle parameterization method [3] and the joint angle parameterization method [4], Xu et al. [5] applied the above two methods to the inverse kinematics solution of a 7-DOF space manipulator, where the arm angle parameterization method has obvious geometric significance and is convenient for arm configuration control, but the method is limited by the robot arm geometric configuration and is not universal.

Inverse Kinematic Algorithm for 8-DOF Redundant Manipulators Based on Weighted Least-Norm method and Parameterization method
The joint angle parameterization method, also called the fixed joint method [6], can be used for the inverse kinematic solution of general redundant robotic arms, because after parameterizing one or more joints, it is equivalent to the reduction of the degrees of freedom of the robotic arm, and then the difficulty of the inverse kinematic solution will be reduced, but it will lose the motion flexibility of the original redundant mechanism itself [7]. Rong et al. [8] performed a kinematic analysis of a large shotcrete robot by converting the joint-coupled motion of an 8-DOF robot arm to kinematic constraints for an inverse kinematic solution, but this inverse kinematic approach was only applicable to the mechanism itself, not to other types of manipulators. Wu et al. [9] used the interval search genetic algorithm based on trajectory to solve inverse kinematics of an 8-DOF shotcrete manipulator, however, this method cannot guarantee the joint angle limitation effectively. Shimizu et al. [10] parametrized the self-motion variable of the SRS (i.e., 3R-1R-3R) 7-DOF anthropomorphic manipulator and obtained the analytical inverse solutions through geometric method. The self-motion of the manipulator is represented by full circumferential rotation around the intersection of the shoulder and wrist joint, but this solution method is only suitable for the SRS manipulator.
The numerical solution method for solving the inverse kinematics is generally based on the inverse Jacobi matrix at the velocity domain, and the main methods are the Jacobi pseudo-inverse method [11,12], the gradient projection method [13][14][15], and the weighted minimum norm method [16][17][18], which approximates the exact solution by giving initial values and iterating. However, the numerical solution method and intelligent algorithms lead to the emergence of cumulative errors, and the computational complexity of the algorithm makes it difficult to ensure real-time performance. To address the shortcomings of the numerical solution method, Yang et al. [19] proposed a method to avoid the inverse of the Jacobi matrix based on the weighted minimum norm method, and the simulation results proved that the method can effectively reduce the solution complexity. Zu et al. [20] proposed a quadratic computational method combining the gradient projection method and the fixed joint method for effectively solving the inverse solution of redundant robots for ensuring the solution accuracy and the real-time performance of the system, which has some universality. Yan et al. [21] solved the inverse kinematics of a 7-DOF redundant manipulator based on a quadratic algorithm. The simulation results show that the algorithm has the advantages of avoiding joint limits and accurate solutions.
In addition, there are some intelligent algorithms based on particle swarm optimization algorithm [22], genetic algorithm [23,24], neural network algorithm [25][26][27], and hybrid biogeography-based optimization algorithm [28] for inverse kinematics solution, but these algorithms generally have problems of poor stability and low applicability, such as slow convergence and prematurity.
Aiming at the inverse kinematics problem of 8-DOF shotcrete manipulator, this paper proposes an inverse kinematics solution method for redundant manipulator based on weighted minimum norm method and joint angle parameterization method, which solves the problems that the shotcrete manipulator cannot obtain closed-form solution, and that the numerical inverse solution algorithm or intelligent algorithm cannot solve the problem of low efficiency and poor stability, At the same time, it retains the redundant characteristics of the manipulator, and has the advantages of avoiding joint limit and no error of closed solution. It is an effective method to solve the inverse kinematics of redundant manipulators.

Forward Kinematics
The robotics research model is an 8-DOF tunnel shotcrete manipulator as shown in Figure1, whose configuration is R-R-P-R-R-P-R-R series mechanism, where R represents the revolute pair and P represents the prismatic pair. The overall structure of the manipulator has divided into three parts: the big arm (joint 1, 2, 3), the forearm (joint 4, 5, 6), and the gunjet (joint 7, 8, and nozzle). Joint 3 and joint 6 are prismatic and the rest revolute. The offset between joint 3 and joint 4 is a4. Joint 5 is a revolute joint of height d5 for yaw motion of the forearm and the offset between joint 4 and joint 5 is a5.   Craig (1986) [29]. This assignment results in the interlink homogeneous transformation matrix where θi denotes the i-th joint angle, sin( ) The corresponding D-H parameters are given in Table 1.

T T T T T T T T T (2)
3 Inverse kinematics optimization algorithm based on weighted least norm method

An optimization method
For a manipulator with redundant degrees of freedom, since the operating space dimension is greater than the joint space dimension, for a given generalized velocity of the end-effector, there are infinitely many groups of solutions to the corresponding joint velocity, and the inverse kinematics formula of the velocity is +( ) is the identity matrix.
It can be seen from Eq. (4) that the gradient projection algorithm only optimizes the secondary task index function () H θ in the Jacobian null space without affecting the main task. However, the gradient projection method has certain drawbacks. For example, the K value is not well chosen [30]. If it is too large, it will cause oscillation and cause unnecessary self-motion. If it is too small, the optimization effect will not be achieved and the algorithm will be unstable.
The weighted least norm method was first proposed by Whitney [31] to solve the redundancy problem. Chan [32] and Dubey used the weighted least norm method to achieve joint limits avoidance for the redundant manipulator. Compared with the gradient projection method, it will not cause unreasonable large self-motion, and will only affect when necessary. The core idea of the algorithm is that when the joint limit is approached, the weighting factor of the corresponding joint increases, thus penalizing the corresponding joint velocity to avoid joint overrun; when there is a tendency to move away from the joint limit or when there is no motion of the joint, no weighting is done to the joint velocity and the joint is free.

Weighted least norm method based on optimization function of avoiding joint limits
The weighted second norm that defines the joint velocity is as follows: Inverse Kinematic Algorithm for 8-DOF Redundant Manipulators Based on Weighted Least-Norm method and Parameterization method is a symmetric positive definite matrix. For simplicity, W is generally taken as a diagonal matrix: (6) where wi, i=1, 2, …, 8 is the weight coefficient. Introduce the following transformations: 11 22 , where Jw and w θ are the weighted Jacobian matrix and the weighted joint velocity vector, respectively. Therefore, the forward velocity kinematics can be expressed as: (8) Then, the minimum norm solution of Eq. (8) can be expressed as: is the pseudo-inverse of the weighted Jacobian matrix. Combined with Eq. (7), the joint velocity solving formula of the weighted minimum norm method can be obtained: The optimization index function H(θ) [33] to achieve the limits avoidance of the joint angle is as follows: where max i  and min i  are the upper and lower limits of the joint angle of the i-th joint, respectively. The partial derivative of H(θ) with respect to θi can be expressed as: When the joint angle limit is approached, / i H   will tend to infinity and the weight coefficient wi will increase to infinity to penalize further motion of the joint; when it is in the middle of the joint angle range, the value of / i H   will be zero, and the joint will be free to move and the associated weight coefficient wi should be equal to 1, given by the following expression: The joint angle 2 () t θ obtained from Eq. (14) is equivalent to that obtained by integrating the velocity over a certain time. The solution has the advantages of avoiding the limit of joint angle and the smallest weighted norm of joint speed.

Inverse kinematics analytical algorithm based on joint angle parameterization method
For the discrete-time real-time motion planning of the redundant manipulator, the joint angles obtained from Eq. (14) are based on the previous moment and obtained by integrating the velocity over a period of time, so the joint angles obtained are a set of optimized approximate solutions, and if a long trajectory is given, the manipulator tip will produce cumulative errors. To improve the accuracy of the inverse solution results, the joint angle parameterization method is used to parameterize the n-m=2 joints of the robotic arm, so that robotic arm becomes a non-redundant mechanism to reduce the difficulty of solving the analytical solution.

Analytical expression of each joint angle variable
Assume that the end pose of the manipulator represented by the homogeneous transformation matrix is given as:  (15) where 0 8 R represents the orientation of the end coordinate system in the base coordinate system, and 0 8 P represents the position of the end coordinate system in the base coordinate system.

Analytical expression for joint angle 1
The forward kinematics equation is left multiplied by the matrix to obtain the following formula:  T  T  T  T  T  T  T  T  T R (16) where L and R are the matrices on both sides of this equation. All elements in the matrix are represented in the appendix.
To simplify the calculation, joint angle 5 and joint angle 6 are parameterized. In the first step, according to Eq. (16), the equation relation between joint 1, joint 5, and joint 6 can be obtained as follows: where, the function of Atan2(Y, X) represents the four-quadrant inverse tangent, which returns the four-quadrant inverse tangent (tan -1 ) of Y and X. There are two values of θ1 obtained, which can be selected according to the actual available range of joint motion. Here, the available range of joint angle 1 is ( , )

 −
, and there are two cases in which the solution results need to be converted to the available range: The interval conversion of the subsequent joint variables is also carried out according to the above method, which will not be described in detail later.

Analytical expression for the sum of joint angle 2 and joint 4
According to Eq. (16), the following equation can be obtained: n c n s a += (29) where n1 and n2 are respectively: Since the sum θ24 of the angles of joint angle 2 and joint angle 4 has been obtained, and θ2 has been obtained above, the value of joint angle 4 can be easily obtained:

Analytical expression for joint angle 8
The analytical expressions for the four joint angles have been derived from the above formulae, leaving only the joint variables θ7 and θ8. Since joint angle 7 and joint angle 8 are only related to the orientation of the manipulator tip, to facilitate the solution of these two joint angles, the following equation is obtained by rearranging the forward kinematic formulae:

T T T T T T T T T R (34)
According to the equality of the corresponding elements on both sides of the matrix, the following equations are obtained: 22

A synthesis algorithm based on weighted least-norm method and parameterization method
The flow chart of the inverse kinematics synthesis algorithm is shown in Figure 4. Firstly, a set of optimized joint variables θ * was obtained based on the weighted least norm method. Further, θ5 and d6 are extracted from the optimized joint variables as the parameterized joint variables. In this case, the manipulator was equivalent to a non-redundant manipulator. Then, based on the joint angle parameterization method, the formulas of the parameterized non-redundant manipulator were deduced to obtain the analytical expressions of the other joint angles.

Numerical results
The known link constants are a4=597mm, a5=330mm, d5=380mm, and the range of joint angles is shown in Table  2. In Table 2, the unit of revolute pair is radians, and the unit of prismatic pair is meter. Table 1 Range of motion of all joint angles

Verify inverse kinematics algorithm
Suppose the initial joint angle is θ0, the corresponding initial pose is X0, and the expected end pose is Xn. where the vector X= [px, py, pz, α, β, γ] T is represented by 6 elements, in which the first three elements represent the position of the end-effector and the last three elements represent the position of the end-effector represented by the Euler angles. The revolute pair and the prismatic pair in the 8-DOF shotcrete manipulator are equivalently represented by cylinders and prisms respectively. The simulation model of the redundant manipulator mechanism established in MATLAB is shown in Figure 5.

Figure 5
The simulation model of manipulator mechanism The tploy quintic polynomial trajectory planning function in the MATLAB robot toolbox is used to interpolate the linear trajectory. The initial and final velocity is set to zero and the simulation time is 5 seconds. Figure 6 shows the comparison curve between the desired Inverse Kinematic Algorithm for 8-DOF Redundant Manipulators Based on Weighted Least-Norm method and Parameterization method trajectory and the actual trajectory obtained by the synthesis algorithm.

Figure 6
The desired trajectory and the actual trajectory The position tracking error of the tip of the manipulator is defined: where ∆x, ∆y, and ∆z are the errors of the actual trajectory and the desired trajectory in the three directions of x, y, and z respectively. The trajectory error curve obtained by the simulation is shown in Figure 7. From the simulation results of Figure 6 and Figure 7, it can be seen that the actual trajectory completely coincides with the desired trajectory, and the position tracking error is zero, which proves the correctness of the inverse kinematics synthesis algorithm.

Inverse kinematics algorithm analysis
The curve of each joint angle obtained by the above linear trajectory simulation is shown in Figure 8 and Figure  9. The configuration changes of the redundant manipulator during the simulation are shown in Figure 10. It can be seen from Figure 8 to Figure 10 that the joint angles change curve is smooth and continuous, and at the same time, the change of the forearm movement is greater than that of the big arm movement. Such motion characteristics can help reduce energy consumption and alleviate the vibration problem of the manipulator.    Figure 11 and Figure 12, and the configuration changes of the manipulator during the movement are shown in Figure 13.

Figure 11
The displacement curves of all revolute joints by the parameterization algorithm As can be seen from Figure 12, the inverse kinematic solution obtained using the parameterization method in this case, the range of motion of the robot arm joint 3 exceeds the joint angle limit and the corresponding motion cannot be achieved in reality. In addition, the parameterization method does not have the joint velocity norm optimization, so it does not have the energy consumption optimization characteristics.

Figure 12
The displacement curves of all prismatic joints by the parameterization algorithm Comparing the configuration changes of the manipulator in Figure 10 and Figure 13, it can be seen that the change in the joint variables of the big arm is greater than that of the forearm., so there is an unreasonable robotic arm movement phenomenon. Therefore, if this method is used alone, the joint angle range of the manipulator may be exceeded, and the situation of inability to move may occur, resulting in the target point becoming unreachable.

Figure 13
The configuration changes of the manipulator during the movement are simulated by the parameterization method Compared with the joint angle parameterization method without optimization characteristics, the inverse kinematic synthesis algorithm can optimize the joint motion by using the redundant characteristics of the mechanism, and the joint variables θ5 and d6 that are parameterized during the parameterization are given by the optimized approximate Inverse Kinematic Algorithm for 8-DOF Redundant Manipulators Based on Weighted Least-Norm method and Parameterization method solutions based on the weighted least norm method, which can effectively avoid the arbitrariness of artificially given values. Comparing the results of these two algorithms, it can be demonstrated that the inverse kinematic synthesis algorithm based on the weighted least norm method and the joint angle parameterization method has certain superiority in the joint limit avoidance and motion optimization of the redundant manipulators.
For the inverse kinematics numerical solutions such as the gradient projection method and the weighted minimum norm method, to ensure the accuracy of the end-effector trajectory, it is necessary to add the pose feedback and iterate the pseudo-inverse of the Jacobi matrix in each control cycle to gradually converge the error to the tolerance interval. The algorithm in this paper only needs to calculate the Jacobi pseudo-inverse once in each sampling period, and there is no iterative process, so the computational complexity is lower, and the high accuracy of the solution can be guaranteed because the analytical expression of the inverse kinematics is obtained.
The processor used for the simulation experiments in this paper is Intel(R) Core(TM) i5-4200U @1.6GHZ, RAM is 8GB, The inverse kinematics algorithm is written in MATLAB, and the inverse kinematics algorithm time consumption experiment is carried out. The linear trajectory is divided into 500 points, and the calculation time of the divided 500 points is averaged, and the average time consumption of the whole inverse kinematics algorithm is 0.7ms per step, among which the average time consumption of the optimization algorithm based on the weighted minimum norm method is 0.45ms, accounting for about 64%, and the average time consumption of the parameterization method is 0.19ms, accounting for about 27%, which fully meets the computational efficiency requirements for general motion control situations.

Conclusions
(1) This study proposes an inverse kinematic synthesis algorithm for the 8-DOF shotcrete manipulator based on the weighted least norm method and the joint angle parameterization method. The algorithm first obtains the optimized approximate solutions for the joint variables based on the weighted least norm method and then obtains the analytical expressions for the remaining joint variables based on the joint angle parameterization method. Finally, the correctness of the algorithm is verified by simulation results. (2) Compared with the traditional inverse kinematics numerical algorithm, the inverse kinematics synthesis algorithm in this paper avoids multiple iterations, has higher computational efficiency, and has no errors in the inverse kinematics results. The weighted minimum norm algorithm is used to give the parameterized values required by the joint angle parameterization method. This algorithm has a certain universality for the inverse kinematics problems of general redundant manipulators. (3) The research results of this paper provide a reference for the realization of intelligent upgrading of tunnel shotcrete manipulator, which has high application value and lays the foundation for further realization of fully automatic spraying of tunnel shotcrete manipulator and modernization of tunnel engineering.