Feedforward control design of flexible robot can take advantage of planning algorithm. In this paper two such algorithms have been presented, one for static case and the other for dynamic case. Both the algorithms are iterative in nature and are computationally based on the minimization of position or tracking error defined at the end-effector level. The algorithms are simple but computationally expensive. The performance is tested on a planar two-link arm through simulation and the results show the capability of the methods to compensate error due to deflection of the flexible links at the time of static condition of positioning and to track a given trajectory effectively at the time of dynamic condition of tracking under end kinematic constraints.

Motion planning under fixed travelling time between start and goal point along a specified path is considered in this study. The structure between the actuator and the end point of the robot arm is assumed flexible and is the main cause of positioning error because of residual vibration. An algorithm is developed to generate motion of the manipulator arm along the specified path that significantly reduces the residual vibration, at the goal point configuration. The problem also considers the full nonlinear dynamics of the manipulator arm. The dynamics are described by finite element method using Lagrange's equation and are expressed by the time scale factor. The performance index is optimized successfully through off-line computation for a two link flexible manipulator, one example of which is described in detail.

The tracking error of robot manipulators based on flexible arm models are presented in this study. The number of actuator is equal to the dimension of the task space which is usually equal to the rigid body degrees of freedom. Actuator torques required for a flexible arm to track a given trajectory are formulated and computed by an iterative algorithm which is based on path planning method. The formulation is complete in the sense that it also considers the full nonlinear dynamics of the manipulator arm. The Euler-Bernoulli beam theory is used to model the dynamics of flexible links and the resulting equations of motion are derived by Lagrange's equation using finite element discretization technique. The algorithm is simple to implement but computationally expensive. The performance and capabilities of the proposed method are tested on a two-link arm through simulation and the results show the potential use of this method not only for feedforward open-loop control but also for closed-loop control strategies.

Two methods are presented to design the joint trajectory for a two-link flexible manipulator: 1) a combined Fourier series and polynomial function; and 2) a B-spline function. In both cases, the path satisfies the start and end kinematic constraints of motion. Using the energy approach for the above two cases, the final path is generated optimizing kinetic and strain energy at the goal point that reduces the residual vibration significantly. The present study is to compare the two methods and to predict that the B-spline curve has faster convergence property and at the same time, minimizes the performance index by six-fold and thus gives further reduction of residual vibration compared to the combined Fourier series and polynomial function. The dynamics of the manipulator arm, during its motion as well as free vibration after reaching the goal, are described by finite element analysis using Lagrange's equation of motion.

Rapid end point positioning along a specified path is considered in this study. The structure between the actuator and the end point of the robot arm is assumed flexible and is the main cause of positioning error because of residual vibration. An algorithm, based on optimization of residual energy at the goal point and the fundamental property of time scaling of manipulator dynamics, is developed that iteratively determines the time optimal trajectory along the specified path under given start and end point conditions. It also considers the full nonlinear dynamics of the manipulator arm and its actuator saturation limit, The dynamics are described by finite element method using Lagrange's equation and are expressed by the time scale factor. The performance index is optimized successfully through off-line computation for a two link flexible manipulator, one example of which is described in detail.