1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

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

16. 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.

17. 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.