Modeling and design optimization of a robot gripper mechanism

https://doi.org/10.1016/j.rcim.2016.12.012Get rights and content

Highlights

  • A general robot modeling and optimal design process is proposed.

  • A case study of a robot gripper is carried out in details to illustrate the proposed process.

  • Data flow and interactions between the geometric, kinematic, and dynamic models are emphasized.

  • A multi-objective optimization design of the gripper is realized using NSGA-II algorithm.

  • A local sensitivity analysis of an optimal solution is performed to identify the most critical links of the gripper.

Abstract

Structure modeling and optimizing are important topics for the design and control of robots. In this paper, we propose a process for modeling robots and optimizing their structure. This process is illustrated via a case study of a robot gripper mechanism that has a closed-loop and a single degree of freedom (DOF) structure. Our aim is to conduct a detailed study of the gripper in order to provide an in-depth step-by-step demonstration of the design process and to illustrate the interactions among its steps. First, geometric model is established to find the relationship between the operational coordinates giving the location of the end-effector and the joint coordinates. Then, equivalent Jacobian matrix is derived to find the kinematic model; and the dynamic model is obtained using Lagrange formulation. Based on these models, a structural multi-objective optimization (MOO) problem is formalised in the static configuration of the gripper. The objective is to determine the optimum force extracted by the robot gripper on the surface of a grasped rigid object under geometrical and functional constraints. The optimization problem of the gripper design is solved using a non-dominated sorting genetic algorithm version II (NSGA-II). The Pareto-optimal solutions are investigated to establish some meaningful relationships between the objective functions and variable values. Finally, design sensitivity analysis is carried out to compute the sensitivity of objective functions with respect to design variables.

Introduction

Robot design is a very complex process involving great modeling and simulation efforts. It has suffered an important progress in the last decades and many approaches deal with this issue. Major steps in robot manipulator design are; kinematics design, dynamics design, thermal design, and stiffness design [1]. In particular, robot modeling and structural analysis are required in all industries. To address these requirements, a design process is proposed in this paper that combines both; robot modeling and geometrical optimization. The proposed process is a sub-process of the general robotics design process in which modeling and optimization activities play essential and complementary roles in the design. As an illustrative case study, we carry out a modeling and an optimal design of a planar single degree of freedom (DOF) mechanism that is used for robot hands or grippers. These kind of mechanisms is amply used because of its simplicity and it only needs one actuator to move it, so many robots use this kind of mechanisms as gripper. However, many researches deal with geometric, kinematic, and dynamic modeling of the robots using different techniques. Some others work on optimization methods for multicriteria robot design optimization. A survey of these research works is presented in the following paragraphs.

Modeling is essential for design specifications, simulation, and advanced control of robots. Different techniques of modeling are available for modeling robots, especially for parallel and closed-loop robots due to their complexity [2], [3], [4], [5]. Ibrahim and Khalil presented kinematic and dynamic modeling of three degrees of freedom 3-RPS (revolute, prismatic, and spherical) parallel robot [6]. This robot is characterized by a coupling between the 6-DOF of the platform. After presenting a (6×3) kinematic Jacobian matrix, they developed a reduced (3×3) Jacobian matrix relating the linear velocity of the platform with respect to the three actuated joints. In another paper, Khalil and Guegan presented closed form solutions for the inverse and direct dynamic models of the Gough-Stewart parallel robot. The models are obtained in terms of the Cartesian dynamic model elements of the legs and of the Newton-Euler equation of the platform [7]. Andrzej et al. used forward and inverse kinematic problem as well as working space and strength analysis issues for the construction of 3-DOF tripod electro-pneumatic parallel manipulator [8]. Qin et al. proposed analytical modeling of a two-staged parallel mechanism composed by a rigid platform in a serial connection with a compliant platform [9]. Hassan and Abomoharam performed a study of a gripper that has two closed loop structure. After finding geometric and kinematic models, they determined the geometrical solution space and verified it via a CAD model of the gripper [10]. Ha et al. employed Hamilton's principle, Lagrange multiplier, geometric constraints, and partitioning method to derive the dynamic equations of a slider-crank mechanism. They showed that dynamic formulation could give a good interpretation of a slider-crank mechanism by comparing the numerical simulations with experimental results [11]. Özgür and Mezouar exploited screw theory expressed via unit dual quaternion representation and its algebra to formulate both the forward (position and velocity) kinematics and pose control of an n-DOF robot arm [12].

Different researches of the optimum design of robot manipulators are available in the works of [13], [14], [15], [16]. Xie et al. proposed a decoupled 3-DOF parallel tool head without parasitic motion. Using the atlases of the tool architecture as bases, the optimal kinematic design of the tool head is carried out [17]. Jiang et al. presented a dynamic modeling and redundant force optimization of a 2-DOF parallel kinematic machine with kinematic redundancy in order to minimize the position errors of the manipulated platform [18]. Nevertheless, in real robot design problems, the number of design parameters can be very large, and their influence on the value to be optimized (the objective function) can be very complicated, having a strongly non-linear character. In these complex cases, stochastic optimization techniques including evolutionary algorithms such as genetic algorithms (GA) may offer solutions to the problem [19]. Coello et al. proposed GA-based multiobjective optimization hybrid technique to optimize the counterweight balancing of a robot arm [20]. Jamwal et al. used a modified genetic algorithm to optimize the kinematic design of a parallel ankle rehabilitation robot [21]. Osyczka and Krenich discussed some new methods for multicriteria design optimization using evolutionary algorithms. The main aims of these methods is to reduce the computing time and to facilitate the decision making process. Examples of a robot gripper mechanism and a clutch break design are presented in this paper showing that these methods can be used to solve different design optimization problems [22]. Gao et al. described the implementation of genetic algorithms and artificial neural networks as an intelligent optimization tool for the dimensional synthesis of the spatial 6-DOF parallel manipulator. The multi-objective optimization (MOO) problem was consisted of two functions: system stiffness and dexterity, which are derived according to kinematic analysis of the parallel mechanism [23].

The rest of the paper is organized as follows. The proposed modeling and optimal design process of the robots and its advantages are described in Section 2. In Section 3, our case study of a robot gripper mechanism is described and its geometric modeling is recalled. Section 4 reviews the kinematic modeling of the gripper, then, the dynamic model is derived in Section 5. After describing and modeling the gripper, the corresponding multi-objective optimization problem is formalized in Section 6. Section 7 describes the solution algorithm of the optimization problem, and the non-dominated sorting genetic algorithm version II (NSGA-II), it discusses the results. Section 8 presents the sensitivity analysis of the gripper mechanism design. Finally, Section 9 summarizes the contributions and results made in this paper and gives some perspectives.

Section snippets

Modeling and optimal design process

The design of robots is a complex engineering task, in which certain mathematical models are required. This task can often be seen as an optimization problem in which the robot parameters or structure describing the best quality design is sought. In this paper, an integrated modeling-optimizing robot design process is proposed where the modeling steps are combined with the optimal structural design process, Fig. 1 illustrates this proposed process. During this procedure, the geometric

Description of the gripper mechanism and geometric modeling

The gripper is planner closed-loop mechanism with a single DOF. The gripping force F, applied on the object, is generated by the actuating force P. Due to the symmetry; we can perform the study on a half of the mechanism that is composed of three links and four joints (one prismatic and three revolute joints), as shown in Fig. 2.

The notations of Khalil and Kleinfinger [25], are used to describe the geometry of the closed-loop structure of the gripper. The definition of the local link frames are

Kinematic modeling

Let η(q1,...,qn)=η(θ,ϕ)=0 be the constraint equations (Eq. (6)) previously found. Taking the time derivative and rearranging these Eq., Eq. 7 can be obtained asK(q)θ̇+K*(q)ϕ̇=0where θ=r2 is the actuated variable (joint) and ϕ=(θ1,θ3,θ4)T are the passive variables (joints). n is the total number of the gripper joints, and m be the number of the actuated joints. Columns of the ((nm)×m) matrix [K(q)] are the partial derivatives of η(q) with respect to the actuated variables θi, i=1,…, m. Columns

Dynamic modeling

To obtain dynamic model of the gripper, Lagrangian formulation is used. As above, n is the total number of joints, and m is the number of the actuated joints. The Lagrangian formulation for a closed-loop mechanism isddt(Lq̇i)Lqi=Qi+j=1nmλjηjqi,i=1,....,nwhere the scalar Lagrangian is defined from the total kinetic and potential energyL(q,q̇)=i=1n(KEiPEi)

KEi kinetic energy of the link i.

PEi potential energy of the link i.

Qi the externally generalised forces applied on link i, it

Optimization problem formulation

The goal of the optimization problem is to find the dimensions of the gripper elements and to optimize objective functions simultaneously by satisfying the geometric and force constraints. The vector of six design variables is x=(d3,d4,d5,l,e,f), where d3,d4,d5,l,e,f are the gripper link variables used in geometrical modeling. The structure of geometrical dependencies of the mechanism is described in Fig. 3. The angles β=π2θ1 and α=βθ3 can be written in terms of the design variables as.

 α=

Solution algorithm and result discussion

After formulating the optimization problem, the next step is to find an optimal design solution using an appropriate algorithm. Our gripper case study is a MOO problem; and all of the above constraints must be taken into account. In order to solve it and find an optimal solution, there are several methods which are proposed in the literature. Due to the complexity, the size of problem and the importance of reducing the solving time, NSGA-II algorithm is used. This algorithm results in Pareto

Design sensitivity analysis

After selecting an optimal design solution, the last step is to analyse the design sensitivity to design variable changes. Design sensitivity analysis computes the rate of objective function change with respect to design variable changes. With the robot structural analysis, the design sensitivity analysis generates a critical information, gradient, for design optimization. Obviously, the objective function is presumed to be a differentiable function of the design, at least in the neighbourhood

Conclusion

This paper has proposed a two-stage process for modeling and optimizing robot structures. In the modeling stage, an analytical step-by-step study is introduced to find geometric, kinematic and dynamic models. The optimization stage shows the procedure to formulize and to solve a design optimization problem and then to analyse the design sensitivity. The data flow and interactions between these steps and stages are highlighted when a robot modeling and optimal design are needed. To illustrate

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