Determining robot’s maximum dynamic load carrying capacity in point-to-point motion by applying limitation of joints’ torque

INTRODUCTION The dynamic load carrying capacity of a robot is defined as “the maximum load that a robotic system can carry provided that the motors' torques do not exceed the saturation limits”. DLCC is one of the criteria for selection of robotic systems. Generally, two methods (direct and indirect) exist for solving the problem of DLCC (1-2). Direct method: This method is based on the discretization of a system’s dynamic variables (state and control variables) that leads to a parameter optimization problem. Then, linear optimization methods (3), nonlinear optimization methods (4), evolutionary techniques (5) or Stochastic Techniques (6) are employed to obtain the optimal values of the parameters. The variables may be classified as state variables, control variables or both (7). The linearizing procedure in direct method and its convergence is a challenging issue, especially when nonlinear terms are large and fluctuating (7, 8). In this way, the obtained answer is an approximate solution which is directly related to the order of polynomial function. Wang et al. (4) have solved an optimal control problem by using ‘Bspline’ functions to calculate the maximum load of a fixed manipulator. The main idea of their research is to discretization the joints trajectories by using Bspline functions and then determining the parameters through nonlinear optimization so as to obtain a local minimum which yields the constraints. A shortcoming of this method is that it limits the solution to a fix-order polynomial (9). Glob. J. Sci. Res., 3 (3): 12-24, 2014 13 | P a g e Iterative Linear Programming (ILP) is another direct method by which a trajectory optimization problem becomes a linear programming problem. The first formulation of this method for calculating the maximum load carried by a simple robot has been provided by Wang and Ravani (4). The linearization operation in the ILP method and its convergence towards the optimal path are difficult tasks, especially when a system has a large degree of freedom or it contains large and fluxionary nonlinear terms. Korayem and Ghariblu used the ILP method to determine the DLCC of a robotic arm with elastic links and also with elastic joints for point-to-point motion and also for motion along a specified trajectory (10, 11). They formulated the DLCC problem as an optimization problem and then employed the ILP method (a direct method) to solve the problem. In their work the boundary conditions are hardly satisfied and there is an almost 10% error in the final solution. Indirect method: This is another method for obtaining the optimal trajectory of the maximum payload. The indirect method, which is based on the PMP, was initially used to solve optimal control problems (12). This method was employed to solve the problems of obtaining the minimum time of motion along specified trajectories (13). In this method, the optimality conditions are extracted as a set of differential equations which, along with the given boundary conditions, form a TPBVP. These sets of differential equations are solved by means of numerical techniques such multiple shooting method (14) or Gradient method (GM) (15). By solving this problem an exact solution can be found. Through this approach, the optimal trajectories for fixed and redundant robots can be calculated by considering different objective functions such as the maximization of the load carrying, minimization of the movement time and minimization of torque, etc. By applying an indirect method which yielded a TPBVP, Korayem and Nikoobin obtained a two-link robot’s DLCC in a point-to-point task (16). In this article, first, a two-link robot’s DLCC is determined by applying torque constraint on the joints (as was previously obtained by Korayem and Nikoobin (16)). Then by revising the method used in Ref. (16) and considering the application of dynamic torque, the problem is resolved and then the obtained results are compared to each other. So, the rest of the paper is organized as follows. In Section 2 the mathematical modeling of the problem will be described. Section 3 is devoted to extract optimality conditions and the TPBVP. In Section 4, first, a two-link robot’s DLCC is obtained using the method applied in Ref. (16), and then this problem is resolved by assuming dynamic torque of each joint. And finally in Section 5 the conclusions from the present work are summarized. Problem formulation: The dynamical model of a robot is described in the Lagrangian formulation as: U q G q q q C q q D    ) ( ) , ( ) (     (1) where U is the torque vector of the joints, D is the inertia matrix, C represents the centripetal and Coriolis forces and G expresses the effects of gravity (2). By using the state vector as:               q q X X X  2 1 (2) In the state space form, Eq. (1) is expressed as: ) , ( U X F X   (3) where F is defined as:                   U X Z X X N X F F F 1 2 1 2 2 1 , (4) ) ( ) ( )] ( ) , ( )[ ( ) , ( 1 1 1 1 2 2 1 1 1 2 1 X D X Z X G X X X C X D X X N  