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Ch09_Controls - Contents Contents 1 Nomenclature 9 Controls...

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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 ff ects . . . . . . . . . . . . . . . . . . . . . . 7 9.4.3 Mechanical advantage . . . . . . . . . . . . . . . . . . . . 8 9.4.4 Armature inertia and speed reduction . . . . . . . . . . . 9 April 7, 2009 17:53
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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 fi 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 ff 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 ff 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
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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 ff 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 to move the mass along a trajectory then generally ¨ x d 6 = 0 and ˙ x d 6 = 0 , although the equation error dynamic still has the same equation form.
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