Dynamics

Manipulator dynamics is concerned with the equation of motion, the way in which the manipulator moves in response to torques applied by the actuators, or external forces. There are two problems related to manipulator dynamics that are important to solve:

• inverse dynamics in which the manipulator's equations of motion are solved for given motion to determine the generalized forces required for each joint (control stage) and
• direct dynamics in which the equations of motion are integrated to determine the generalized coordinate response to applied generalized forces (simulation stage).

The equation of motion for an n-axes manipulator are given by

 

Where

q is the vector of generalized joint coordinates describing the pose of the manipulator

is the vector of joint velocities

is the vector of joint accelerations

M is the symmetric joint-space inertia matrix, or manipulator inertia tensor

C describes Coriolis and centripetal effects

F describes viscous and Coulomb friction and is not generally considered part of rigid- body dynamics

G is the gravity loading

Q is the vector of generalized forces associated with generalized coordinates q

The equation may be derived via a number of techniques, including the Lagrangian method. Due to the enormous computational cost of this approach it is always difficult to compute manipulator torques for real-time control based on the dynamic equations. To achieve real-time performance many approaches were suggested, including table lookup and approximation [4]. The most common approximation is to ignore the velocity-dependent term C, since accurate positioning and high speed motion are exclusive in typical robot application. Practically, a PID controller might be a good option to achieve a real-time performance,

 

 
Figure 3: PID control loop  

where , and are the derivative, proportional and integral parameters respectively. See Figure 3 for a schematic of a PID control loop.
The advantages of using a PID controler are the following:

• Simple to implement
• Suitable for a real-time control
• The behavior of the system can be controlled by changing the feedback gains