An Improved Weighted Gradient Projection Method for Inverse Kinematics of Redundant Surgical Manipulators

Different from traditional redundant manipulators, the redundant manipulators used in the surgical environment require the end effector (EE) to have high pose (position and orientation) accuracy to ensure the smooth progress of the operation. When analyzing the inverse kinematics (IK) of traditional redundant manipulators, gradient-projection method (GPM) and weighted least-norm (WLN) method are commonly used methods to avoid joint position limits. However, for the traditional GPM and WLN method, when joints are close to their limits, they stop moving, which greatly reduces the accuracy of the IK solution. When robotic manipulators enter a singular region, although traditional damped least-squares (DLS) algorithms are used to handle singularities effectively, motion errors of the EE will be introduced. Furthermore, selecting singular region through trial and error may cause some joint velocities exceed their corresponding limits. More importantly, traditional DLS algorithms cannot guide robotic manipulators away from singular regions. Inspired by the merits of GPM, WLN, and DLS methods, an improved weighted gradient projection method (IWGPM) is proposed to solve the IK problem of redundant manipulators used in the surgical environment with avoiding joint position limits and singularities. The weighted matrix of the WLN method and the damping factor of the DLS algorithm have been improved, and a joint limit repulsive potential field function and singular repulsive potential field function belong to the null space are introduced to completely keep joints away from the damping interval and redundant manipulators away from the unsafe region. To verify the validity of the proposed IWGPM, simulations on a 7 degree of freedom (DOF) redundant manipulator used in laparoscopic surgery indicate that the proposed method can not only achieve higher accuracy IK solution but also avoid joint position limits and singularities effectively by comparing them with the results of the traditional GPM and WLN method, respectively. Furthermore, based on the proposed IWGPM, simulation tests in two cases show that joint position limits have a great impact on the orientation accuracy, and singular potential energy function has a great impact on the position accuracy.

Real-Time Flow Optimization of Hydraulic Manipulator with One Degree of Redundancy Considering Joint Limit Constraint

Due to the complexity in unstructured environments (e.g., rescue response and forestry logging), more hydraulic manipulators are equipped with one redundant joint to improve their motion flexibility. In addition to considering joint limit constraint and maneuverability optimization like electrically driven manipulators, hydraulic manipulators can optimize flow consumption consider flow optimization aiming at energy saving and flow anti-saturation for redundancy resolution, since multiple joints are supplied by one pump. Therefore, this paper proposes a redundancy resolution method combining the gradient projection method with a weighted Jacobian matrix (GPM-WJM) for real-time flow optimization of the hydraulic manipulator with one degree of redundancy considering joint limit constraint. Its solution consists of two parts: a special solution (the weighted least-norm solution) and a general solution (the projection of the optimization index in the null space of the weighted task Jacobian matrix). Simulations are carried out to verify its effectiveness. The simulation result shows that GPM-WJM can meet the constraints of joint limit without affecting the tool center point (TCP) trajectory and utilize the remaining redundancy to optimize the flow consumption and manipulability in real-time, which can reduce average system flow by 10.45%. Compared with the gradient projection method (GPM) for flow optimization, GPM-WJM can reduce the maximum acceleration when avoiding the joint limits by 80% at the cost of slightly weakening the flow optimization effect, which is beneficial to improve the accuracy of the manipulator in practice.

APPLICATION OF GRADIENT PROJECTION METHOD TO PARAMETRIC OPTIMIZATION OF STEEL LATTICE PORTAL FRAME

The paper has proposed a mathematical model for parametric optimization problem of the steel lattice portal frame. The design variable vector includes geometrical parameters of the structure (node coordinates), as well as cross-sectional dimensions of the structural members. The system of constraints covers load-carrying capacities constraints formulated for all design sections of structural members of the steel structure subjected to all ultimate load case combinations. The displacements constraints formulated for the specifiednodes of the steel structure subjected to all serviceability load case combinations have been also included into the system of constraints. Additional requirements in the form of constraints on lower and upper values of the design variables, constraints on permissible minimal thicknesses, constraints on permissible maximum diameter-to-thickness ratio for the structural members with circle hollow sections, as well as the conditions for designing gusset-less welded joints between structural members with circle hollow sections have been also considered in the scope of the mathematical model. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used to solve the formulated parametric optimization problem. New optimal layouts of the steel lattice portal frame by the criterion of the minimum weight, as well as minimum costs on manufacturing and erection have been presented.