Dynamics of Manipulators. Newton-Euler and Lagrangian Solution Approach

Introduction to Dynamics of Manipulators

Dynamics is the study of forces and the Dynamics of Manipulators is the study of forces associated with manipulators. To perform a task by a manipulator its members and joints have to be moved. To move something we need force. And to move something in a particular way forces and torques have to be applied in a certain way to obtain the desired trajectory.

In Dynamics of Manipulators we study forces applied to manipulators. To perform a particular task the manipulator is accelerated from rest to start moving, then the end-effector may be required to be moved with a constant velocity and then decelerated to bring it to rest at the desired point. Such motion requires variation of torques at the joints by actuators in accordance to the desired trajectory.

Our task in Dynamics of Manipulators is to find the torque to be generated by the torque actuators at the manipulator joints. The functions of the torque variation depend upon the trajectory to be followed by the manipulator, masses of links, friction in link joints and force applied by or payload at the end-effector.

Problems to Be Solved

Dynamic analysis of manipulator has two types of problems to be solved.

1. The trajectory with variation of position, velocity and acceleration is given and torques required at manipulator joints to move along the desired trajectory are to be found.
2. Torques variations are given and the motion of manipulator has to be found. It may involve finding position, velocity and also acceleration.

The position, velocity and acceleration here also denote the respective angular quantities.

Dynamic Equations of Motion

Dynamic Equations of Motion relate position, velocity and acceleration of links of the manipulator to the forces applied to its links. As the links of manipulators are three dimensional bodies with defined dimensions, the dynamic analysis have to consider their orientation, angular velocity and angular acceleration. To take into consideration the effect of mass distribution of links inertia tensor of the links is used in the analysis.

To derive the dynamic equations of motion of manipulators two types of methods can be followed

Newton-Euler Method

In the dynamic analysis we are provided with the acceleration, linear as well as angular, and we have to find the required force or torque. Newton’s equation relates mass distribution and acceleration to find force and Euler’s equation does the same for angular quantities. In this method we start from a link and move on to another links one by one transforming acceleration and then return through same path finding forces and torques from one link to another.

Lagrangian Method

The Lagrangian Method is based on the energy. In this method we define a quantity called as the Lagrangian. Lagrangian is kinetic energy of the manipulator minus its potential energy. This lagrangian is used for deriving the dynamic equations of motion of a manipulator. Equations obtained are same as those obtained from Newton-Euler Method.

An important use of the dynamic equations of motion developed for the manipulator operation is in simulation of manipulators. The equations are reformulated to give position, velocity and acceleration as a function of actuator torques. From this we can simulate the manner in which the manipulator will move under application of given torques.

This post is part of the series: Mechanics of Robotic Manipulators

Robotics is perhaps the outcome of the desire to synthesize such machines which can perform tasks like humans do and even beyond, that is, performing task which humans cannot do, and that too in a way even better than what humans do. This series introduces you to Mechanics of Robotic Manipulators.