Denavit-Hartenberg Research Articles

A Neuro-Fuzzy Solution for Inverse Kinematics of Redundant Manipulators

Abstract: This paper presents a novel approach for solving the inverse kinematics (IK) problem of redundant robotic manipulators using an Adaptive Neuro-Fuzzy Inference System (ANFIS). Redundant manipulators, characterized by having more degrees of freedom (DOF) than necessary for a task, introduce complexity in their IK solutions due to their non-linear, time-varying, and transcendental nature. Traditional methods such as algebraic or geometric solutions are often computationally intensive and can suffer from issues such as singularities. This study leverages ANFIS, combining the learning capability of neural networks with the logic-based structure of fuzzy systems, to predict IK solutions for 5-DOF and 7-DOF manipulators. The Denavit-Hartenberg (D-H) notation is employed for the kinematic modeling of robot links. Comparative analysis of the predicted IK values using ANFIS demonstrates its high accuracy and efficiency, as validated by surface and residual plots. The results show that ANFIS can be effectively utilized for fast and reliable IK predictions, offering a robust alternative to traditional approaches. This method can be applied in robotic systems requiring high precision in constrained or dynamic environments.

Treatment Couch Path Planning for Proton Therapy Systems

In the treatment process of proton radiation therapy, the patient needs to be positioned and immobilized before being moved into the treatment position. In this study, the patient was primarily positioned using the 6R robotic treatment couch as the patient support system (PSS). A simplified three-dimensional model of the treatment room was developed based on the relative motion within the treatment room. The forward and inverse kinematics of the 6R robotic treatment couch were analyzed using an improved Denavit-Hartenberg (D-H) representation. A collision interference model was created based on the actual treatment process. The motion path of the treatment couch was planned and simulated in MATLAB using an improved artificial potential field method for obstacle avoidance. The results indicate that the robotic treatment couch can smoothly navigate around obstacles to reach the target point, satisfying the positioning requirements for proton therapy.

Research on Omnidirectional Gait Switching and Attitude Control in Hexapod Robots

To tackle the challenges of poor stability during real-time random gait switching and precise trajectory control for hexapod robots under limited stride and steering conditions, a novel real-time replanning gait switching control strategy based on an omnidirectional gait and fuzzy inference is proposed, along with an attitude control method based on the single-neuron adaptive proportional–integral–derivative (PID). To start, a kinematic model of a hexapod robot was developed through the Denavit–Hartenberg (D-H) kinematics analysis, linking joint movement parameters to the end foot’s endpoint pose, which formed the foundation for designing various gaits, including omnidirectional and compound gaits. Incorporating an omnidirectional gait could effectively resolve the challenge of precise trajectory control for the hexapod robot under limited stride and steering conditions. Next, a real-time replanning gait switching strategy based on an omnidirectional gait and fuzzy inference was introduced to tackle the issue of significant impacts and low stability encountered during gait transitions. Finally, in view of further enhancing the stability of the hexapod robot, an attitude adjustment algorithm based on the single-neuron adaptive PID was presented. Extensive experiments confirmed the effectiveness of this approach. The results show that our approach enabled the robot to switch gaits seamlessly in real time, effectively addressing the challenge of precise trajectory control under limited stride and steering conditions; moreover, it significantly improved the hexapod robot’s dynamic stability during its motion, enabling it to adapt to complex and changing environments.

Manufacture of a 4-Degree-of-Freedom Robot to Support an IRB 120 Robot

In this work, we present the construction and control of a four-degrees-of-freedom (DOF) manipulator aimed at addressing one of the key challenges faced by the Academy-Industry Cooperation Center (CCAI): the need for mechatronic equipment to support and facilitate the development of advanced robotic cells. We begin by designing the robot’s structure and components using SolidWorks software for computer-aided design (CAD) modeling. This ensures that all the links and parts fit together properly without collisions. The robot links are then manufactured using 3D printing. Additionally, we performed kinematic modeling, dynamic analysis, and PI-V control, along with control using a trigonometric function (hyperbolic tangent). To evaluate the robot’s movement, we simulate these processes using Matlab R2019a/Simulink software, focusing on key parameters such as position, velocity, and acceleration, which inform the design of PI-V control for each link. We also present the electrical and electronic designs, followed by system implementation. The kinematics of the robot play a crucial role in the dynamics and controller design. We validate the kinematics using Peter Corke’s libraries based on the Denavit–Hartenberg parameters. The results show that the controller based on the trigonometric function improves the response time, particularly enhancing the performance of axes 2 and 3.

Dynamic simulation and adaptive heave compensation control of a wave-compensated gangway under random wave excitation

For high-precision position control of wave-compensated gangways under random wave excitation, the kinematic model is established based on the Denavit–Hartenberg parametric method. Desired motion parameters for each degree of freedom (DOF) are solved through multi-DOF motion decoupling calculations. The telescopic arm, pitching arm, and slewing seat are simplified as mechanical-hydraulic motor-driven systems or cylinder-driven systems. The nonlinear dynamic models for each system are deduced based on their structural composition and adaptive motion controllers are designed with the desired motion parameters as control targets. The gangway base's heave motion under random wave excitation is reconstructed and the electromechanical-hydraulic coupling dynamic model of the wave-compensated gangway under random wave excitation is coupled. Comparative testing with the traditional Proportional-Integral-Derivative control method shows that the displacement stabilization accuracy of the telescopic arm's tip in the x-axis and y-axis directions under the condition of scale 4 wave improved by 93.05% and 97.38%, respectively. This research provides a technical reference and reliable approach for dynamic analysis and high-precision control of wave-compensated gangways.

A trajectory tracking control method for the discharge arm of the self-propelled forage harvester

The cooperative operation between the self-propelled forage harvester and the trailer aims to achieve precise and automatic forage unloading. The discharge arm serves as the core structure for conveying forage, and the precision and speed of its motion control are crucial factors influencing the efficiency of forage harvesting. In this context, we proposed a trajectory tracking control method for the discharge arm based on an improved particle swarm optimization (IPSO)-PID controller. Firstly, we utilized the Denavit-Hartenberg (D-H) model of the discharge arm for a positive kinematics solution and the geometric resolution and linear fitting method for the inverse kinematics solution. Secondly, we employed the polynomial interpolation method for trajectory planning on the joint space of the discharge arm, and the PSO algorithm for time-optimal trajectory planning. Then, we designed an IPSO-PID trajectory tracking control algorithm and built an Amesim-Simulink co-simulation model for multiple simulation experiments. Finally, we conducted several performance tests of the discharge arm automatic control system in the workstation and the field, respectively. The experiment results indicate that the performance of the IPSO-PID controller exceeds that of all other controllers, which can meet the discharge arm’s motion control accuracy and speed requirements. Our research results are of great significance for improving the productivity and automation process of the self-propelled forage harvester and provide valuable references for research on automatic and precise control of material loading in other agricultural cooperative harvesting.

Application of the Water-Based Electro-Hydraulic Actuator (EHA) to the Heavy-Duty Collaborative Robot

In this paper, the design of a driving mechanism for a heavy-duty collaborative robot (cobot) capable of lifting payloads up to 20 kg is presented. This study focuses on an articulated robot utilizing a water-based Electro-Hydraulic Actuator (EHA). The Denavit–Hartenberg (D–H) representation was employed to relate the rotational angles and the end-effector’s location, facilitating the design of the actuators. The maximum required torques for joints 2 and 3, responsible for lifting for 12 s, were calculated under quasi-static and dynamic loading conditions. The results showed that the maximum required torques were 126.67 Nm and 58.86 Nm for joint 2 and 3, respectively. The maximum torque for joint 2 occurs when the pitch links are fully extended, whereas the maximum torque for joint 3 occurs when the third link is parallel to the ground. The torques, due to the inertia and Coriolis dynamic terms, were also calculated and found to be lower than those required for the gravitational term. Various maneuvering scenarios, along with Ansys Motion simulation, were analyzed for the verification of the results. Based on the calculated maximum torques, the linear actuators of the EHA were designed. The heavy-duty cobot can be built with the developed actuator proposed in this paper. The total weight of the entire frame was measured to be 14.59 kg, resulting in a high Payload/Weight (P/W) ratio of 1.37. In conclusion, the robot was made lighter and can operate more efficiently, particularly for heavy loads up to 20 kg.

A Magnetic Climbing Robot for Steel Bridge Inspection

Abstract. The maintenance of traditional steel bridges presents many significant challenges due to the fact that bridges of different ages and standards vary in design, structural complexity and technical difficulty. In light of these considerations, it is clear that maintenance personnel must possess a wealth of professional expertise and experience in order to effectively address the diverse and intricate maintenance challenges that arise. Moreover, the maintenance process entails high-risk operations such as high-altitude work and welding, which pose a threat to the safety of construction personnel. Therefore, this study introduces a magnetic adsorption robot designed to address the low efficiency and high costs of traditional maintenance methods. This robot features a dual robotic arm mechanism and modeled using the Denavit Hartenberg (D-H) parameter method. Electromagnetic adsorption is utilized to achieve wall adhesion, and direct power supply is provided through DC power to control weight. The robot can perform flexible remote control operations through the HC-05 Bluetooth module. The experimental results show that the robot exhibits robust adhesion and obstacle crossing ability on steel surfaces, thereby enabling unimpeded movement within the steel structure. In addition, the robot is now capable of performing overhaul procedures via remote control.

Study of Collaborative Space in Rebar Tying Robotic Systems

Abstract In the process of rebar tying operations conducted by multiple robots, the flexibility and operability of the robot’s end effector directly influence the overall performance of the rebar tying system. This reflects the crucial role of the coordinated working capability of multiple robots in the trajectory planning of the rebar tying system. Utilizing an eight degrees of freedom rebar tying system composed of two collaborative robots and a truss structure, the D-H (Denavit-Hartenberg) dynamic model for each robotic arm is established. The working area of the system is ascertained via Monte Carlo methods. Flexibility and operability are adopted as metrics to evaluate the system’s performance, subsequently leading to the formulation of a mathematical model for assessing the system’s flexibility. MATLAB is employed to delineate the robotic working range and simulate analysis of flexibility and operability when operating on the rebar tying plane, yielding an intuitive distribution of system flexibility. By analysing the distribution of flexibility and operability across the working plane at varying robot spacing intervals, an optimal robot movement distance can be determined. This ensures that during operational tasks, the rebar tying system not only possesses a larger total working area but also maintains a high level of flexibility and operability within the collaborative region, thereby fulfilling the objective of enhancing workspace utilization and improving rebar tying efficiency.

Trajectory planning and automatic docking of LNG five-axis loading arm

PurposeIn response to the challenges posed by the conventional manual flange docking method in the LNG (Liquefied Natural Gas) loading process, such as low positioning accuracy, constraints on production efficiency and safety hazards, this study analyzed the LNG five-axis loading arm’s main functions and structural characteristics.Design/methodology/approachAn automated solution for the joints of the LNG loading arm was designed. The forward kinematic model of the LNG loading arm was established using the Denavit–Hartenberg (D-H) parameter method, and its workspace was analyzed. The Newton–Raphson iteration method was employed to solve the inverse kinematics of the LNG loading arm, facilitating trajectory planning. The relationship between the target position and the joint variables was established to verify the stability of the arm’s motion. Flange center identification was achieved using the Hough transform function. Based on the ROS platform, combined with Gazebo and Rviz, an experimental simulation of automatic docking of the LNG loading arm was conducted.FindingsThe docking errors in the XYZ directions were all less than 0.8 mm, meeting the required docking accuracy. Moreover, the motion performance of the loading arm during docking was smooth and free of abrupt changes, validating its capability to accomplish the automatic docking task.Originality/valueThe proposed trajectory planning and automatic docking scheme can be used for the rapid filling of LNG filling arms and LNG tankers to improve the efficiency of LNG transportation. In guiding the docking, the proposed automatic docking scheme is an accurate and efficient way to improve safety.

Robot error compensation strategy based on error sensitivity

Abstract Due to the errors in manufacturing and assembly, there are differences between the actual model and the theoretical model of the robot, which affects the positioning accuracy of the robot end-effector. In order to improve the accuracy of robot end-effector position, a robot error compensation strategy based on error sensitivity is proposed.Firstly, the robot kinematic model is established by Denavit-Hartenberg (D-H) method, and the sensitivity of end-effector position error is analyzed. According to the influence degree of different kinematic parameters on the robot end-effector position accuracy in the whole workspace, different weights are given to different kinematic parameters. {\color{red}Secondly, the kinematic error model is established, and the redundancy of the error parameter matrix is analyzed to obtain an independent error model. Thirdly, based on the error sensitivity analysis, a Weighted Levenberg-Marquard (WLM) algorithm with adaptive penalty factor is proposed, and the kinematic parameters are iteratively identified.} Finally, an error compensation experiment is carried out by using a universal serial six-degree-of-freedom robot. The experimental results show that the maximum error, mean absolute error and root mean square error of the position error on the test set are reduced by 90.75%, 89.86% and 95.64% respectively. The research in this paper provides a theoretical basis for robot end error compensation.

The Utility of Magnetic Resonance Imaging-based Vertebral Bone Quality Scores as a Predictor of Cage Subsidence Following Transforaminal and Posterior Lumbar Interbody Fusion.

Retrospective study. The objectives were to determine whether vertebral bone quality (VBQ) scores are associated with interbody cage subsidence following transforaminal (TLIF) and posterior (PLIF) lumbar interbody fusions and whether there is a clinically sensitive threshold for subsidence. Interbody cage subsidence following lumbar fusion is a complication that can generate poor surgical outcomes. Prior research has correlated cage subsidence with bone mineral density. VBQ scores derived from magnetic resonance imaging (MRI) have been proposed as a tool for measuring bone mineral density, offering a potential new and convenient preoperative risk assessment tool for subsidence. The study involved patients undergoing single-level PLIF or TLIF between 2007 and 2022. Exclusions were for nondegenerative diagnoses, multilevel/revision surgeries, inadequate radiographs, missing immediate postoperative radiographs, and preoperative MRI studies older than 1 year. VBQ was calculated at L1-L4 from preoperative T1-weighted MRI images. Subsidence was assessed by changes in disc height (DH; >2mm difference) and segmental lordosis (SL; >5 degrees difference) between immediate weight-bearing postoperative and latest postoperative lateral radiographs. Statistical analysis included descriptive and inferential statistics. Subsidence was observed in 27% (SL parameter) and 47% (DH parameter) of 51 total patients. VBQ scores were significantly associated with cage subsidence based on both SL (odds ratio = 7.750, P = 0.012; correlation coefficient = 0.382, P = 0.006) and DH (odds ratio = 4.074, P = 0.026; correlation coefficient = 0.258, P = 0.057) in the combined TLIF/PLIF cohorts. In the cohort of 36 patients undergoing TLIF, a VBQ of 2.70 yielded 100.0% sensitivity and 46.2% specificity in detecting subsidence with SL measurement (area under the curve = 0.812, P < 0.001) and 86.7% sensitivity and 47.6% specificity with the DH measurement (area under the curve = 0.692, P = 0.033). We found that MRI-based VBQ scores are effective predictors of cage subsidence following TLIF surgery. A VBQ score of 2.70 demonstrated a reliable model and high sensitivity for doing so, identifying a potential clinical threshold for preoperative subsidence risk assessment. Level III.

Calibration of rotary axis angular positioning deviations in a six-axis robotic manipulator by using the R-Test

Prediction accuracy of the kinematic model plays a pivotal role in offline numerical compensation for three-dimensional (3D) positioning error of a six-axis robotic manipulator. In addition to the Denavit-Hartenberg (D-H) error that was typically discussed in the analogous research, this paper proposes a novel scheme to identify the bidirectional angular positioning deviations of all the rotary axes without using the commercial laser tracker. This proposed scheme employs the R-Test, which is significantly more cost-efficient than a laser tracker employed in many past works. Compared to the conventional D-H model, significantly improved prediction performance of the proposed model was experimentally investigated on three rectangular paths covering the entire workspace. Experimental results indicate that the proposed scheme reduces the residual error from 505.2 to 215.2 mm, compared to 118.8 mm with a laser tracker, while maintaining a significantly lower cost that is 1/100 of a commercial laser tracker. Uncertainty analysis also clarify that when only the D-H error parameters are identified, as in many conventional works, the angular positioning deviations can be a major uncertainty contributor to the identification of the D-H errors. The proposed scheme does not require the R-Test locations to be calibrated, and its influence on the uncertainty in the identified errors is assessed.

Research on designing dynamic model of TITAN-VIII quadruped robot based on the robot structure

Quadruped robots have many advantages over wheeled robots in that they have high mobility and can move on any terrain. However, the control problem of four-legged robots will be more complicated because the robot's kinematic model has a high order and the dynamic process is complicated, so it is necessary to implement order reduction methods when constructing kinematic equations for four-legged robots. The article proposes a simplified method to determine the kinematic equations for four-legged robots called TITAN-VIII. The Denavit-Hartenberg (D-H) rule is used to analyze the forward kinematics. The inverse kinematic equations are determined on the basis of mathematics and the structure of the robot's legs. The research results are simulated on MATLAB Simulink software.

Gait planning based on bionic quadruped robot

Based on the ankylosaurus, a four-legged robot structure with 14 degrees of freedom was designed in this study. The kinematic model was established using the Denavit-Hartenberg (D-H) method. The inverse kinematics of the robot was analyzed, and the angle equations of each leg joint were obtained. In order to reduce the kinetic energy generated when leaving the ground and landing, the compound cycloid was modified. The modified foot curve effectively reduces energy and meets the kinematic requirements. On the basis of the foot trajectory, the diagonal leg phase difference was set to 0.5, and the diagonal gait was adopted as the gait of the bionic quadrupled robot. Simulink software was used to construct the simulation environment, and the maximum error of the simulation results and the theoretical step height was 2.05%, and the step length maximum error was 2.39%. During walking, the thigh joint angle was −92.82° to −80.21°, and the hip joint angle was 22.23° to 43.83°. Compared with the simulation trajectory and the theoretical curve, because the experiment did not consider the balance of the fuselage, there was a certain error when the leg was raised to the highest point. Overall, the gait planning strategy designed in this study basically achieves the expected effect, lays a certain foundation for the practical application of the bionic quadruped dinosaur robot, and also provides a reference for the subsequent research.

Direct Closed-Loop Control Structure for the Three-Axis Satcom-on-the-Move Antenna

The traditional Satcom-on-the-Move (SOTM) mechanical structure consists of a dual-axis configuration with an azimuth axis and a pitch axis. In this structure, when the pitch angle is 90 degrees, the rotation of the azimuth axis cannot change the antenna’s direction. To solve this issue, a three-axis SOTM mechanical structure has been developed. The traditional three-axis SOTM servo control system adopts a closed-loop control scheme. In this scheme, due to the difficulty in directly obtaining the antenna’s rotation angle, the angles of rotation for each axis are typically selected to represent the antenna’s rotation angle. The closed-loop feedback includes the angles and angular velocities of the axes, which cannot completely capture the antenna’s motion state, essentially constituting an indirect closed-loop control. Addressing the shortcomings of this indirect closed-loop control, this paper first establishes the kinematic relations between the axes of the three-axis SOTM antenna using the Denavit–Hartenberg (DH) method. Subsequently, the relationship between antenna pointing and the rotational states of the three axes was derived using the Jacobian operator. Building upon this foundation, a direct closed-loop control structure for a three-axis SOTM antenna was designed. To enable the control system to achieve rapid convergence with minimal overshoot, an Active Disturbance Rejection Control (ADRC) algorithm based on smooth continuous functions is introduced as the inner and outer loop controller algorithms within the direct closed-loop control structure. To address the nonlinearity in the design scheme, a piecewise linearization method is proposed to reduce the demands on the microprocessor’s performance and enhance the engineering feasibility of the solution. Finally, the effectiveness of the proposed approach is validated through experiments. The experimental results demonstrate that compared to traditional indirect closed-loop control methods, utilizing the direct closed-loop control method for the three-axis SOTM antenna presented in this paper can lead to higher precision in pointing the antenna towards satellites and enhance communication effectiveness.

Research on the Optimal Trajectory Planning Method for the Dual-Attitude Adjustment Mechanism Based on an Improved Multi-Objective Salp Swarm Algorithm

In this study, an optimization method for the motion trajectory of attitude actuators was investigated in order to improve assembly efficiency in the automatic docking process of large components. The self-developed dual-attitude adjustment mechanism (2-PPPR) is used as the research object, and the structure is symmetrical. Based on the modified Denavit–Hartenberg (MDH) parameter description method, a kinematic model of the attitude mechanism is established, and its end trajectory is parametrically expressed using a five-order B-spline curve. Based on the constraints of the dynamics and kinematics of the dual-posture mechanism, the total posturing time, the degree of urgency of each joint, and the degree of difficulty of the mechanism’s posturing are selected as the optimization objectives. The Lévy flight and Cauchy variation algorithms are introduced into the salp swarm algorithm (SSA) to solve the parameters of the multi-objective trajectory optimization model. By combining the evaluation method of the multi-objective average optimal solution, the optimal trajectory of the dual-tuning mechanism and the motion trajectory of each joint are obtained. The simulation and experiment results show that the trajectory planning method proposed in this paper is effective and feasible and can ensure that the large-part dual-posture mechanism can complete the automatic docking task smoothly and efficiently.

Human-Exoskeleton Coupling Simulation for Lifting Tasks with Shoulder, Spine, and Knee-Joint Powered Exoskeletons.

In this study, we introduce a two-dimensional (2D) human skeletal model coupled with knee, spine, and shoulder exoskeletons. The primary purpose of this model is to predict the optimal lifting motion and provide torque support from the exoskeleton through the utilization of inverse dynamics optimization. The kinematics and dynamics of the human model are expressed using the Denavit-Hartenberg (DH) representation. The lifting optimization formulation integrates the electromechanical dynamics of the DC motors in the exoskeletons of the knee, spine, and shoulder. The design variables for this study include human joint angle profiles and exoskeleton motor current profiles. The optimization objective is to minimize the squared normalized human joint torques, subject to physical and task-specific lifting constraints. We solve this optimization problem using the gradient-based optimizer SNOPT. Our results include a comparison of predicted human joint angle profiles, joint torque profiles, and ground reaction force (GRF) profiles between lifting tasks with and without exoskeleton assistance. We also explore various combinations of exoskeletons for the knee, spine, and shoulder. By resolving the lifting optimization problems, we designed the optimal torques for the exoskeletons located at the knee, spine, and shoulder. It was found that the support from the exoskeletons substantially lowers the torque levels in human joints. Additionally, we conducted experiments only on the knee exoskeleton. Experimental data indicated that using the knee exoskeleton decreases the muscle activation peaks by 35.00%, 10.03%, 22.12%, 30.14%, 16.77%, and 25.71% for muscles of the erector spinae, latissimus dorsi, vastus medialis, vastus lateralis, rectus femoris, and biceps femoris, respectively.

Design and kinematics analysis of a parabolic cylinder deployment mechanism with modular over-constrained triangular pyramid

Space missions require novel mechanisms that can have compact form in complex space environments. This study proposes a modular triangular pyramid parabolic cylinder deployment mechanism. The modular deployable unit contains an over-constrained triangular pyramid deployable mechanism and a symmetrical trapezoidal Bricard linkage that drives the longitudinal and transverse motions of the mechanism. A cable net is added between two symmetrical linkages to form a parabolic cylindrical reflector and is analyzed by the force density method. The Denavit-Hartenberg (D-H) coordinate method is used to analyze the kinematic characteristics. Based on the analytical solution for the angular displacement, the degrees of freedom of the mechanism are derived from screw theory. Subsequently, the angular velocity and acceleration of the joint points are obtained. Finally, kinematic models of the modular triangular pyramid parabolic cylinder deployable mechanism are established, and the accuracy of the theoretical model is verified using a numerical method. This novel parabolic cylinder deployable mechanism will have a significant influence in aerospace domain. Crucially, the work has value to design deployment mechanism for parabolic cylinder antenna.

Simultaneous compensation of geometric and compliance errors for robotics with consideration of variable payload effects

In order to satisfy the high accuracy requirements of robotic applications, it is necessary to consider not only the geometric errors but also the compliance errors which are caused by the self-gravity of the link and the external payloads. A general error model is developed based on the modified Denavit-Hartenberg (DH) model. For the parameters in the error model, there is a coupling between the compliance coefficients and the link parameters, making it difficult to use only the compliance coefficients to compute the compliance errors due to external payloads. As the external payload for the robot manipulator varies, the parameter identification of the model should be conducted again. In this paper, a novel algorithm using a variable payload method is proposed to first identify the compliance coefficients using different payloads and end effector position information. Second, the kinematic parameter errors and link parameters are identified with the given compliance coefficients. Then, the algorithm generates a modified trajectory using the calibrated DH tables for the precision compensation. Simulation and experimental results demonstrate that the positioning accuracy can be improved by 80 % to 90 % even under different payloads. The root mean square, mean, maximum, and standard deviation of the residual errors by using the proposed algorithm could outperform the conventional kinematic algorithm.