Ch09_Controls

Ch09_Controls - Contents Contents i 1 Nomenclature i 9...

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Unformatted text preview: Contents Contents i 1 Nomenclature i 9 Controls 1 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9.2 Position control of a 1DOF mass . . . . . . . . . . . . . . . . . . 1 9.3 Position control of a 1DOF mass with control partitioning . . . . 2 9.3.1 Addition of integral control . . . . . . . . . . . . . . . . . 4 9.4 Joint control with control partitioning . . . . . . . . . . . . . . . 5 9.4.1 Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.4.2 Unmodeled e f ects . . . . . . . . . . . . . . . . . . . . . . 7 9.4.3 Mechanical advantage . . . . . . . . . . . . . . . . . . . . 8 9.4.4 Armature inertia and speed reduction . . . . . . . . . . . 9 April 7, 2009 17:53 Chapter 9 Controls 9.1 Introduction A one degree-of-freedom system is used to illustrate compensation by control law partitioning which is also known as inverse dynamics control. This simpli f es or even eliminates the plant dynamics using a model—based control law so it is then easy to design a servo control law for a desired response. When applied to a multiple input, multiple output system such as a robot, it also has a decoupling e f ect so each joint can be controlled independently. However, this requires a perfect model of the robot so in practice parameter errors and unmodelled e f ects must be considered which couple the joint dynamics. 9.2 Position control of a 1DOF mass A simple one degree-of-freedom mass is used as the plant to illustrate servo control, Figure 9.1. The dynamic model of the plant is f = m ¨ x The input is the action f and the output is the motion ¨ x as well as ˙ x and x . It is desired to move the mass to a desired position x d by regulating f so that x → x d . The relationship that produces f is called the control law. Action f is applied to the mass using actuation. A very common position control law has the form, f = − k v ˙ x + k p ( x d − x ) where k v is the velocity gain and k p is the position gain which are design pa- rameters. Combining the model and control law gives the system dynamics as m ¨ x + k v ˙ x + k p x = k p x d April 7, 2009 17:53 2 Chapter 9 Controls On the left-hand side, the simple mass has been augmented with a virtual damping term k v ˙ x and a virtual spring term k p x . On the right side the force input has been replaced by the desired input scaled by k p . This is often referred to as a servo system with the block diagram illustrated by Figure 9.2. Details such as actuation and sensing are not explicitly shown. The error is the di f erence between the desired and actual positions e = x d − x If the mass is to come to rest at x d then ¨ x d = ˙ x d = 0 and the system can be written in terms of the error dynamic as m (0 − ¨ x ) + k v (0 − ˙ x ) + k p ( x d − x ) = 0 or simply m ¨ e + k v ˙ e + k p e = 0 For an underdamped system the time response x to a step input x d is shown in Figure 9.3 and the error response is shown in Figure 9.4. If it is also desired toFigure 9....
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This note was uploaded on 01/24/2010 for the course ME 6407 taught by Professor Staff during the Spring '08 term at Georgia Tech.

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Ch09_Controls - Contents Contents i 1 Nomenclature i 9...

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