[Abstract]: This paper explores the optimization calculation of a jointed bionic finger with the main goal of reproducing the fist-clenching posture and motion trajectory. The paper focuses on the establishment of the motion matrix, the structural simulation and instructional description of the bionic finger, and presents the sample formula of the expected target, the overall optimization objective F(x), and the constraint functions g[sub]1[/sub](x)～g[sub]42[/sub](x). The approximate solution of the mathematical model is obtained, and its approximation degree with the expected target curve is over 95%, indicating that the optimization calculation achieves good bionic effects. [Keywords]: Optimization objective; Mathematical model; Constraint function. Many industrial robots use jointless end effectors for gripping, hanging, and attracting, thus preventing them from performing fist-clenching movements to grasp objects, and even limiting their overall application. If a bionic finger could be matched to the arm mechanism of an industrial robot, extending with the arm to the grasping point to grasp objects, the robot's complete grasping function would be realized. This paper discusses the optimization calculation of such a bionic finger mechanism. 1. Establishment of the motion matrix Based on the skeletal structure and motor function of the human arm and fingers, the "arm-finger" mechanism of the robot is constructed using 7 links and 6 "5 types" of kinematic pairs. The mechanism (Figure I) is mounted on a 6-DOF robot. The lower link L0 is connected to the body A0, and the end of the higher link 6 is the fingertip "s". The "5 types" of kinematic pairs connected to the links respectively constitute the shoulder joint A[sub]1[/sub], elbow joint A[sub]2[/sub], wrist joint A[sub]3[/sub]... up to the distal joint A[sub]6[/sub] of the finger. The mechanism is an open chain mechanism with 7 links connected in series and has 6 degrees of freedom. The geometric parameters and variable relationships between the coordinate systems of two adjacent links can be represented by homogeneous transformation as: (1) Where L Let be the distance between the centerlines of joints A_i and A_i+1 at both ends of link L; d is the distance between the intersection points of adjacent links L_i-1 and L_i with the centerline of joint A_i, which is the angle between adjacent links L_i-1 and L_i in the plane perpendicular to the centerline of joint A_i. If A_i is a translating joint, then d = 0. The motion of the robot's "arm-finger mechanism" can be described by a motion matrix composed of homogeneous transformations between adjacent link coordinate systems. 3. The motion matrix of the coordinate system of link L with index z relative to the coordinate system of the base link L[sub]i[/sub] is: When the robot walks to its position, A[sub]u[/sub] and L[sub]0[/sub] are considered as fixed parts. If the arm moves to its position and then the finger grasping action begins, link L should be considered as a fixed part again, and the motion matrix of the fingertip "s" is: Where 2 is the expected target acquisition formula . Bionic fingers need to imitate the movement posture of human hand fingers when making a fist and approximate its fingertip trajectory. Therefore, it is necessary to sample human hand fingers and formulate the expected target. 2.1 The posture angle correlation formula shows that when the fingers grasp an object and clench their fist, the motion of each finger is basically the same, and the finger positions are almost on the same curved surface. There is no need to describe the relative motion of the joints and establish a coordinate system at each point. Here, we only take the middle finger as an example and set the origin of the coordinate system on the proximal joint A, as shown in Figure 2. The lengths of the proximal, middle, and distal joints of the finger are , , and , respectively. The postures of the three joints are used to describe the finger. In normal people, the restraint of muscles and tendons during the clenching of the fist cannot be regarded as an independent variable without perception. They must be correlated according to certain rules. Through high-speed photography and sampling measurement of human hand fingers, the posture angle correlation formula of the middle finger for different hand shapes is obtained as follows: where is the critical posture angle; are the lower and upper limits of , respectively, which are determined according to the range of finger activity; C is the additional value of the supercritical angle; and [IMG]/uploadpic/THESIS/2008/3/20080319174651968022E.jpg[/IMG] is the correlation coefficient of the posture angle. List the above values ​​in the sampling table (see Table 1), with the length unit being mm. Expand or shrink the table values ​​as needed and obtain the "enlarged" and "micro" finger data after plotting correction. 2.2 Coordinate formula of the expected target After obtaining the sampling table, different hand shapes and their corresponding data should be selected for grasping objects of different sizes, so as to determine the expected target of the bionic finger. From Figure 2, the coordinate of the middle finger tip "G" is obtained. When the hand shape is selected, the coordinate of the finger tip "G" is only a function of . By changing the value of , the expected target curve can be plotted according to formula (9) and formula (10) as shown in Figure 3. If the trajectory of the finger tip "G" calculated by the bionic finger coincides with the expected target curve, then the ideal bionic degree is achieved. 3 Simulation structure and condition formula of the bionic finger The posture angle θ3 of the distal phalanx of the bionic finger should be close to the value determined by formula (6) and formula (7); the motion trajectory of its finger tip "s" should be close to the trajectory of the finger tip "G" described by formula (9) and formula (10). To achieve the biomimetic requirements of the expected attitude angle and fingertip trajectory, the biomimetic finger should be a jointed finger structure that maintains an inwardly curved posture. 3.1 Joint Simulation The simulated finger mechanism is a three-joint finger mechanism HJ composed of two 3-bar linkages and two closed-loop four-bar linkages (ABCD and CDFG), as shown in Figure 4. The four-bar linkages are all in a crossed state, forming two crossed loops. The first cross ring ABCD simulates the proximal phalanx of the finger. The second cross ring (zFB) simulates the middle phalanx of the finger. The extension of the connecting rod EF, FS, simulates the distal phalanx of the finger. "S" is the end of the finger. A, C, and F are its three joints. 3.2 Inward bending posture condition Only when the two four-bar linkages of the bionic finger always cross without turning outward can the finger present a bionic inward bending posture, and make the fingertip "s" reproduce the same trajectory when the finger is clenched or relaxed. The condition for ensuring the state angle that meets this requirement is: In equation (11), the posture angle of the proximal phalanx AC is obtained when the first closed ring ABCD is in the cross position. Therefore, it is necessary to perform coordinate analysis on the cross position and non-cross position of the ring, and discard the non-cross position by comparison and discrimination method. According to Figure 4, the coordinate analysis of ABCD starts from AB, let the AB rod be the active member, input, and perform coordinate analysis on its possible positions. According to the vector triangles ADB and DBC, the posture angles of the two possible positions of the AC rod are obtained respectively. After simple geometric derivation, they are expressed as: The calculated values ​​are compared in the formula. When the mechanism is in a non-crossing state, the bionic variables should be modified to meet the crossing requirements. In formula (11), the state angle of the distal phalanx CF is obtained when the second closed loop C'EFG is in the crossing position. Similarly, the attitude angles of the two possible positions are derived by the vector triangle: Similarly, and are compared with , and the larger angle value is taken as . 4 Optimization objectives and constraint functions 4.1 Overall optimization objective (1) Finger variable x. Through the analysis of the finger mechanism and motion. The main dimensions and attitude angles that affect the bionic effect are shown in Figure 4. x is (2) Trajectory optimization objective F[sub]1[/sub](x). If the trajectory of the bionic finger tip "S" is close to the expected target curve, the coordinate difference between the "S" point and the "G" point should be minimized (18). In order to reduce the interval error and improve the clarity of the observation of the fist clenching process, the number of calculation points N should be appropriately increased. (3) Posture optimization target F[sub]2[/sub](x) To make the distal phalanx of the designed finger meet the posture requirements of the finger clenching action, its posture angle should be close to the value determined by equations (6) and (7). That is, the optimization target should be determined according to the minimum difference between the two. (4) Optimization target F[sub]3[/sub](x) for the lightest structural weight. In order to make the structure of the bionic finger the lightest, especially for the enlarged finger, its bar length should be minimized. That is, the weighting factor. Its value can be predetermined and then gradually adjusted according to the optimization procedure until the target is balanced and optimized. Finally determined. 4.2 Constraint function g(x) (J) Boundary constraint g(x)～g[sub]8[/sub](x) —— Based on the measurement and statistics of the middle finger joint of a normal medium-sized finger, the constraint function for determining the range of values ​​of the bionic finger variable is (2) Structural constraint g[sub]9[/sub](x)～g[sub]40[/sub](x) respectively restricts the range of values ​​of the structural angles of the crank DCE, BCG, and GFC, so that the four-bar linkages ABCD and CEFG become closed loops and always maintain a cross state: (3) Posture constraint g[sub]4l[/sub](x) and g[sub]42[/sub](x) are constraint functions that make the finger always in an inwardly curved posture during the entire forward and reverse movement: 5 Optimization calculation of mathematical model The mathematical model for the optimization calculation of bionic finger mechanism is 6 Conclusion The optimization calculation method discussed in this paper is not limited to the medium-sized middle finger, but can be extended to "micro" and "amplified" fingers: it is not limited to the clenching action of the fingers, as long as it is combined with the movement of the arm, elbow, and wrist mechanism, it can be extended to the dynamic capture of the active target, and can even be applied to the realization of coordinated arm and finger movements.