The analysis routines related to manipulator motion analysis are inverse kinematics, joint space trajectory generation, and dynamics。 These are evaluated using standard robotic analysis algorithms [21]。
• Inverse kinematics evaluates the required joint variables 8,erai and 8r。or based on the desired Cartesian coordinates of the initial and final positions。
Design Variables From GA/DE
Inverse Pict: & Plzce position
FIGURE 2 Analysis flow chart。
Joint space trajectory generates the position 8(t) (cubic polynomial profile), velocity 8'(/) and acceleration 8”(t) profiles based on the desired motion time and joint variables。
Dynamics evaluates the inpidual joint toi ques based on the structural chin acteris- tics of the links, the payload and the position, velocity and acceleration profiles。 The evaluation of the dynamic equations could include also friction losses proportional to the magnitude of the velocity。 In this work, the friction losses are not considered in the analysis since the friction coefficients are not known and their influence will be to increase the value of the too que requiied for the motion。 The dynamics one eval- uated according to the Newton—Euler method [21]。
(6)
Finite Element Analysis for Deflection
The deflection analysis evalti:ites the deflection of the end effector of the manipulator considering the structural cli:u acteristics of the links, the mateiial properties foi inertia evaluation and the payload。 This calculation is performed using the finite element method [22]。 The maximum deflection occurs when the manipulator is at its maximum reach。 The generalized load vector is derived using the paylodd Fp d the properties of the links of the manipulator such as cross-section, modulus of elasticity, E, and density, p。
The links of the manipulator are modeled as beams with the general 3-D beam element stiffness matrix [22]。 The structural model of a two link planar manipulator is shown in Fig。 3 where the degrees of freedom for each node are the moments, M, due to bending about the c-axis and forces due to axial loading along the y-axis。
The assembled global load vector, P, for the two-element manipulator after reduction
due to motion constraints at degrees of freedom Mb and f is given by
where J, is the link weight per unit length given by f —— (p,N,L,g)/ L,
FIGURE 3 Forces and moments for structure l analysis。
The stiffness matrices for each element are assembled to generate the global stiffness matrix。 The assembled global stiffness matrix for the two-element manipulator, K, after reduction due to constraint degrees of freedom is given by
The deflection vector v = {y , 8 , y , 8 } is then evaluated according to
U —— K —'P (9)
3。 OPTIMIZATION APPROACHES
Three evolutionary based optimization approaches were investigated, a SGA, simple GAE and DE。 机器人优化设计英文文献和中文翻译(4):http://www.youerw.com/fanyi/lunwen_82103.html