marvin:ecp2
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marvin:ecp2 [2009/01/29 10:07] – deva | marvin:ecp2 [2009/01/29 11:02] (current) – rieper | ||
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- | =====Our Angle of Approach Towards Control Theory===== | + | ===== Theory ===== |
+ | ====Our Angle of Approach Towards Control Theory==== | ||
- | Basically there are three approaches to our control problem, which we will describe briefly in an increasing order of complexity. | + | Basically there are three approaches to our control problem, which we will describe briefly in an increasing order of complexity. |
- | The second approach is again to consider the control plant as a black box, but here we wish to estimate the characteristic transfer function of the plant. Recall that if we give an impulse to a filter we are given the filter characteristic by means of the impulse response. The analogy to control theory is to apply a step function and observe the transient response, which is illustrated on the figures in the following section | + | The second approach is again to consider the control plant as a black box, but here we wish to estimate the characteristic transfer function of the plant. Recall that if we give an impulse to a filter we are given the filter characteristic by means of the impulse response. The analogy to control theory is to apply a step function and observe the transient response, which is illustrated on the figures in the following section |
- | The third method is to derive a mathematical model of the dynamics of the control plant and this can be somewhat complex. In the case of a balancing robot it is helpful to look at the mathematical modelling of an inverted pendulum and there next to add the physical dimensions of the balancing robot. When working with modelling we end up with non linear equations, which is a problem as our controllers are linear. One common method is to use the state space representation and then to make a linearization | + | The third method is to derive a mathematical model of the dynamics of the control plant and this can be somewhat complex. In the case of a balancing robot it is helpful to look at the mathematical modelling of an inverted pendulum and there next to add the physical dimensions of the balancing robot. When working with modelling we end up with non linear equations, which is a problem as our controllers are linear. One common method is to use the state space representation and then to make a linearisation |
- | =====Introducing the PID Controller===== | + | ====Introducing the PID Controller==== |
The digital implementation of a PID controller is actually based on very simple filtering techniques similar to what we described in the [[http:// | The digital implementation of a PID controller is actually based on very simple filtering techniques similar to what we described in the [[http:// | ||
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The above pictures are from (([[http:// | The above pictures are from (([[http:// | ||
- | By inspection we summarize the following. An increase in the proportional gain will create a larger overshoot, but reduce the steady state error and rise time. An increase in the integral controller also increase the overshoot, while eliminating the steady state error. If we increase the differential controller, we decrease the overshoot with no affect on the steady state error, but the bandwith | + | By inspection we summarize the following. An increase in the proportional gain will create a larger overshoot, but reduce the steady state error and rise time. An increase in the integral controller also increase the overshoot, while eliminating the steady state error. If we increase the differential controller, we decrease the overshoot with no affect on the steady state error, but the bandwidth |
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</ | </ | ||
- | Further readings and inspiration are found at " | + | Further readings and inspiration are found at this homepage(([[http:// |
- | ===== Defining the States in the Control Loop ===== | + | ==== Defining the States in the Control Loop ==== |
Now that we have established our choice of control loop, we must determine an error function. In a state space representation this corresponds to determining the states, which are defined through differential equations describing the dynamics of the control plant. Therefore we may turn to the model described in the documentation of Rich Chi Ooi((Balancing a Two-Wheeled Autonomous Robot, Author Rich Chi Ooi, The University of Western Australia | Now that we have established our choice of control loop, we must determine an error function. In a state space representation this corresponds to determining the states, which are defined through differential equations describing the dynamics of the control plant. Therefore we may turn to the model described in the documentation of Rich Chi Ooi((Balancing a Two-Wheeled Autonomous Robot, Author Rich Chi Ooi, The University of Western Australia | ||
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{{ : | {{ : | ||
- | This makes perfectly sense, since we need to drive the wheels in a direction that keeps the upper body of the robot in equilibrium. We therefore need sensor readings regarding the position of the wheels and the upper body in order to keep the wheels under the robot’s | + | This makes perfectly sense, since we need to drive the wheels in a direction that keeps the upper body of the robot in equilibrium. We therefore need sensor readings regarding the position of the wheels and the upper body in order to keep the wheels under the robot’s |
{{ : | {{ : | ||
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- | ===== Parameters and Stability | + | ==== Parameters and Stability ==== |
- | The calculation of the right PID control parameters is essential to secure a stable system. This is called loop tuning. If loop tuning is not done according to the task at hand, the control system will become unstable i.e. the output will diverge. Oscillation might occur in this case. The only thing that will prevent | + | The calculation of the right PID control parameters is essential to secure a stable system. This is called loop tuning. If loop tuning is not done according to the task at hand, the control system will become unstable i.e. the output will diverge. Oscillation might occur in this case. The only thing that will prevent |
Loop tuning could be defined as adjusting the control loop parameters (Proportional gain(P), Integral gain (I) and Derivative gain(D)) to the optimum value for a desired control response. The z-transform for the controller should, in order to be stable obey the rules for a stable system, that is poles of the system should be located inside the unity circle, see diagram below. | Loop tuning could be defined as adjusting the control loop parameters (Proportional gain(P), Integral gain (I) and Derivative gain(D)) to the optimum value for a desired control response. The z-transform for the controller should, in order to be stable obey the rules for a stable system, that is poles of the system should be located inside the unity circle, see diagram below. | ||
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There are a number of different approaches when tuning PID parameters of which three will be described below. | There are a number of different approaches when tuning PID parameters of which three will be described below. | ||
- | The methods that can be used range from no knowledge of the system to a full simulation of the system that can be used to calculate stability and set the exact parameters of the PID controller for optimum performance. One thing to note before talking about parameter tuning is the difference between online tuning and offline tuning. Online tuning means tuning when the system is running. That means one adjust parameters while the system is running. Offline tuning means take the control out of the plant, adjust the parameters and then re-engage the system. It is the later method we have used. The robot have been stopped, the parameters have en adjusted in the software, and lastly new firmware have been uploaded. Online tuning could have been used with a bluetooth | + | The methods that can be used range from no knowledge of the system to a full simulation of the system that can be used to calculate stability and set the exact parameters of the PID controller for optimum performance. One thing to note before talking about parameter tuning is the difference between online tuning and offline tuning. Online tuning means tuning when the system is running. That means one adjust parameters while the system is running. Offline tuning means take the control out of the plant, adjust the parameters and then re-engage the system. It is the later method we have used. The robot have been stopped, the parameters have en adjusted in the software, and lastly new firmware have been uploaded. Online tuning could have been used with a Bluetooth |
- | === Manual | + | === Manual |
This is a trial and error approach. You do not need to have any idea about the plant. The procedure is as follows: | This is a trial and error approach. You do not need to have any idea about the plant. The procedure is as follows: | ||
* Set I and D values to zero | * Set I and D values to zero | ||
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There exist different kinds of tuning software. One of these could be (([[http:// | There exist different kinds of tuning software. One of these could be (([[http:// | ||
- | === Summarize of approaches | + | === Summarize of Approaches |
The three different approaches can be seen from the table below. In that table the advantages an disadvantages can me seen together with what knowledge is required. | The three different approaches can be seen from the table below. In that table the advantages an disadvantages can me seen together with what knowledge is required. | ||
^ Method ^ Advantages ^ Disadvantage ^ Requirements ^ | ^ Method ^ Advantages ^ Disadvantage ^ Requirements ^ | ||
^ Manual Tuning | No math required (Online*) | Inaccurate | ^ Manual Tuning | No math required (Online*) | Inaccurate | ||
- | ^ Ziegler-Nichols | Proven method (Online*)| Trial and error, pretty | + | ^ Ziegler-Nichols | Proven method (Online*)| Trial and error, pretty |
^ Software Tools | Consistent tuning, Online or offline tuning, Allow for simulation beforehand | Expensive, knowledge about plant and development tools are required | Mathematical model | | ^ Software Tools | Consistent tuning, Online or offline tuning, Allow for simulation beforehand | Expensive, knowledge about plant and development tools are required | Mathematical model | | ||
* Online can in some cases be a disadvantage, | * Online can in some cases be a disadvantage, | ||
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- | ===== Results | + | ===== Implementation |
First we present the software implementation, | First we present the software implementation, | ||
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This class contains all the PID magic happening that makes Marvin keep its balance. | This class contains all the PID magic happening that makes Marvin keep its balance. | ||
- | The balancing module reads the angles and the angle velocities from the gyro and motors, and uses these values (weighted) to feed to the PID control calculation which again produces the new power to apply to the motors. | + | The balancing module reads the angles and the angle velocities from the gyroscope |
These are the constants used as weights in PID control, and the error calculation: | These are the constants used as weights in PID control, and the error calculation: | ||
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velocities) < | velocities) < | ||
- | The sleep call at the end of the loop, reflects the max sample speed of the Gyroscope, which should be is 300 times per second, 3.33msec between calls, but since the code itself takes up some time, the actual sleep value of 3msec seem to work appropriately. | + | The sleep call at the end of the loop, reflects the max sample speed of the gyroscope, which should be is 300 times per second, 3.33msec between calls, but since the code itself takes up some time, the actual sleep value of 3msec seem to work appropriately. |
The controls are added, in two places: | The controls are added, in two places: | ||
- | * To make Marvin move forward and backwards we add the tilt offset to the gyro angle (at the '' | + | * To make Marvin move forward and backwards we add the tilt offset to the gyroscope |
* To make Marvin rotate wee add the left and right motor offsets directly to the right and left motor power (at the '' | * To make Marvin rotate wee add the left and right motor offsets directly to the right and left motor power (at the '' | ||
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public void run() | public void run() | ||
+ | |||
{ | { | ||
MotorControl motors = new MotorControl(Motor.C, | MotorControl motors = new MotorControl(Motor.C, |
marvin/ecp2.1233220060.txt.gz · Last modified: 2009/01/29 10:07 by deva