Robotics II Copyright Martin P. Aalund, Ph.D. Dynamics of Robots Dynamics give the relationship between the robot’s position (and its derivatives) and forces. What are the Forces Can use the Forces to Calculate Currents Can be used to generate a feed-forward term for a control model Can use forces for typical trajectory to select motors, and design the robots structure. Inverse Given robots desired hand Position, Velocity and Acceleration what are the required joint torques Forward Given the vector of joint torque what will the velocity of the end-effector be. Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Dynamics End Effector Force Trajectory End Effector Position Trajectory Inverse Jacobian Forward Dynamics Forward Kinematics Tool Forces Joint Torques Joint Positions Tool Position Time Time Forward Jacobian Inverse Dynamics Inverse Kinematics Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Methods Newtonian Force Balance Calculate velocity and acceleration and use to get forces F=mA Lagrangian Energy Based Calculate potential and kinetic energy of robot. Use this to determine forces directly. Closed Form Recursive Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Acceleration How do we get the Accelerations of the joints. Linear Acceleration of a vector Q as seen from frame A Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Inertia Inertia is the tendency of a body to remain in a state of rest or uniform motion. A force is required to change this state Motion could be linear or rotational Linear inertia is known as mass The frame in which the inertia is measured is known as the inertial reference frame Center of Mass Point in a body that moves in the same way that a single particl subject to the same forces would. Robotics II Copyright Martin P. Aalund, Ph.D.

Moment of Inertia and Mass A an object has a mass m. The mass is distributed throughout the body. For a single degree of freedom we can speak of just a mass for linear motion. For rotation the equivalent term known as the mass moment of inertia A body has three mass moments of inertia Ixx Iyy Izz and 3 mass products of inertia Ixy, Ixz, Iyz Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Moment of Inertia The mass moments and mass products depend on the shape and mass distribution of the body It is convenient to create a matrix that represents the inertia of a body. It is known as the inertia tensor. We can select a reference frame such that the products of inertia are 0. In such case the moments of inertia are known as the principal moments and the axes as the principal axes. Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Moment of Inertia Inertia around an axes other than the center of mass can be calculate by moving the the center of mass with the parallel axes theorem. Mass moments of simple objects can be combined to create complex objects. Example Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Newtonian Method First iterate out the serial change of the robot starting at link 1 and continuing to I-1. Calculate the linear and angular position, velocity and accelerations of the center mass of each link. Once the accelerations and velocities of the center of mass are calculated we can use the Newton Euler equations to calculate the forces on the center of mass of each link. Once we have the forces we can work back from the last link to calculate the joint forces and torque's. Robotics II Copyright Martin P. Aalund, Ph.D.

Newtonian Method Continued We can include gravitational loads by giving frame 0 an acceleration of on G. This fictitious acceleration causes the same reaction forces and torques at the joints as gravity without additional computational requirements Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. State Space Instantaneous Position, Velocity, and acceleration Can be for each joint Can be for Cartesian coordinates Important for advanced control Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Lagrangian Dynamics Kinetic Energy: Work required to bring a body to rest Potential Energy: Energy due to the removal of an object from the origen of its reference frame (Gravitational, Spring) Function of position, whose negative derivative is force Mechanical energy Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Langrangian Dynamics The Langrangian is defined by We can get the torque of force of joint n from The kinetic energy of a link is: The kinetic energy of a manipulator is: Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Langrangian Dynamics Potential energy of link i is: The manipulators potential energy is: In general the torques of a manipulator are t is a vector of torques Robotics II Copyright Martin P. Aalund, Ph.D.

Simple example of both methods Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Manipulator Dynamics Joint Space M is the Mass Matrix V has two elements Coriolis functions of two different velocities Centrifugal function of the velocity squared G Contains the Gravity Terms All are function of position Cartesian Space Can convert back to Joint Space Robotics II Copyright Martin P. Aalund, Ph.D.

Robotics II Copyright Martin P. Aalund, Ph.D. Which method to use Computationally Newton-Euler is better Lagrangian gives more info Robotics II Copyright Martin P. Aalund, Ph.D.