Optimization of the Roomba Vacuum

Transcription

1 Optimization of the Roomba Vacuum By Donhoon Lee Marc Zawislak ME Winter 006 Final Report ABSTRACT In 00 Roomba was introduced to the public to help time constraint consumers in the vacuuming their houses. The Roomba vacuums the carpet of your house without any assistance using memory and sensors as it travels. By completing an ancient idea of allowing a robot to complete household chores this product has won several design awards. However, since the novelty of the design has left many aspects to improve which our optimization model investigates. The Roomba consists of a combination of mechanical and electrical devices. Several motors control the wheels and drive the vacuum where a battery controls the electrical power into this system. The objective of our project is to optimize the design of Roomba to clean most efficiently before the battery dies by changing the parameters of this machine, for instance, the suction power of vacuum, the size of vacuum inlet area, the radius of driving wheel, the speed of each motor and so on. To accomplish this goal the system is decomposed into the vacuum suction efficiency and the power train. Several tradeoffs and linking variables exist in both the subsystems.

2 Table of Contents 1. Introduction 3 Nomenclature 4. Subsystem Design.1 Vacuum Subsystem Problem statement 7 Mathematical Model 8 Model Analysis 16 Scaling 18 Optimization study 18 Numerical result 0 Parametric study Discussion of result. Driving Wheel Subsystem Problem statement 4 Mathematical Model 4 Model Analysis 34 Optimization Study 37 Parametric Study 39 Discussion of Results System Integration References 45

3 1. Introduction The Roomba was introduced to the public to help time constrained consumers in the vacuuming their houses. Using sensors, the Roomba vacuums the carpet of your house without any assistance. By completing an ancient idea of allowing a robot to complete household chores this product has won several design awards. However, since the design was only brought into the market three years ago there are there are many aspects to improve. The Roomba consists of a combination of mechanical and electrical devices. Several motors control the wheels and drive the vacuum. A battery in the Roomba controls the electrical power into this system. The objective of our project to optimize the design of Roomba to clean the most area before the battery dies by changing the size of housing, speed of body, size of battery and so on. A diagram of our subsystems of the Roomba can be seen in Figure 1. Three values are used to calculate the objective function of cleaning the most area before the battery dies: the area of the vacuum, the distance traveled, and the dirt picked up per distance traveled rate. The area and speed of the vacuum are calculated in one subsystem with respect to the suction power of the motor. The other subsystem calculates the battery life of the vacuum with respect to the size of the battery, motors and housing area. Figure 1. Subsystems of Roomba Optimization The first subsystem attempts to maximize the efficiency of the vacuum area. This area consists of a fan which attempts to create a pressure to draw up the dirt particles and an agitator which sweeps the particles into the air. Both the fan and agitator have motors to drive them which are connected to the main battery source. The vacuum opening, speed of agitator, and flow rate of the fan are the design variables. Several tradeoffs exist in the subsystem which makes it a very interesting problem. 3

4 The second subsystem attempts to maximize the efficiency of the battery used by the power train. The Roomba s driving system consists of the motorized wheels which drive the vacuum to three different motions: straight forward, a right hand turn, or a left hand turn. Motors control both the wheels which need current from the battery. Several tradeoffs exist in the system which makes this optimization a very interesting problem There are several links in-between the two subsystems which could lead to improving the designs independently will not lead us to an overall optimal system design. First, the motor which controls the vacuum and battery would be in two different subsystems. However, the motor dictates how long the battery can last, thus they are linked. Second, the vacuum area would be calculated in a different subsystem than the overall housing size and travel distance of the Roomba. Thus, these dimensions could conflict as the vacuum area cannot be bigger than the overall housing size. In addition, it is inconclusive if a larger vacuum which can travel a shorter distance covers less of an overall area than a vacuum which is smaller but can travel a longer distance before the battery dies. At a minimum, two tradeoffs in the system exist which makes this a very interesting design optimization problem. First, if you would like to maximize the vacuum area that would create a large housing unit which consumes a large amount of power and cannot travel very far before the battery dies. Second, if the suction power and speed of the vacuum is increased the battery life would decrease. A balance between the housing, motors, and battery size will be determined through the design optimization. It will be interesting to see how the actual size of the Roomba compares to the size of the design optimization. Nomenclature The nomenclature used in our model can be seen in Table 1. Nomenclature Symbol Name N Number of Passes P Ambient Pressure ρ Density of the Air l Length of Opening w Width of Opening S A Speed of Agitator D R Diameter of Roomba R A Radius of Agitator P 1 Vacuum Pressure v B Velocity of Air F v Force of Vacuum dirt r Dirt to Area Ratio A A Vacuum Area Covered by Agitator dirt Amount of Dirt in specific area 4

5 ε T i A i V i MAX i T V R A κ ba κ ma κ fa A F A F Flow V A V Flow MAX κ VA κ VF κ A i VMAX R agit Perc flap ρ a μ k gear Agit R h ε A ε B ε V N int i SSV i SSD i sysmax IA DSYS IA VSYS a b R m L V emf J I i K m Total Efficiency Current used by Agitator Current used by Vacuum Maximum current used Total Current Used Voltage of the System Resistance of Agitator Circuit Back-EMF constant of Agitator Motor Torque constant of Agitator Motor Speed Constant of Agitator Motor Torque of Agitator Motor Force of Agitator Total Force Flow Rate of the Fan Area of the Fan Maximum Flow Rate of Fan Vacuum to Agitator Efficiency Coefficient Vacuum to Flapper Efficiency Coefficient Agitator Efficiency Coefficient Maximum Current for Fan Agitator Ratio Percentage to Flapper Density of the Agitator Kinetic Friction Constant Gear Ratio on the Agitator Motor Ratio of the Agitator to Flapper Height of Agitator to Ground Efficiency of Agitator Efficiency of Vacuum to Agitator Efficiency of Vacuum to Flapper Number of Intervals Current used by Vacuum Subsystem Current used by Driving Subsystem Maximum Current used by System Driving Subsystem Current Allocated Vacuum Subsystem Current Allocated Velocity function variable Velocity function variable Terminal resistance in a motor Rotor inductance in a motor Back-EMF voltage Rotor inertia of the motor Moment of inertia of the Roomba s body Current Torque Constant 5

6 K b Back-EMF constant M Moment M 1 Mass of the Roomba s body θ Rotational angle ω Angular velocity ω n Angular velocity of a motor with no load ω Angular acceleration K f Rotational friction constant Torque generated by a motor Torque applied outside s Stall torque w Torque on a wheel ω n No-load Angular velocity i i Start-up current i o No-load current R Gear ratio of a gearbox Required angular velocity Angular velocity of a motor F Force applied to the body d Distance of movement l Distance between two wheels Desired velocity for each pattern motor applied N required N motor v i Table 1. Nomenclature used in Vacuum Subsystem 6

7 . Subsystem Design Subsystem 1: Vacuum Subsystem.1.1 Problem Statement The Vacuum Subsystem calculates how much dirt can be picked up in a given area from the vacuum. This subsystem calculates this by determining the suction force of the vacuum fan and contact force of the agitator. The force of the agitator is used to determine the efficiency of the number of particles swept up. A percentage of the particles are pushed to the rubber flapper and the remaining particles are stuck on the brush. Next, the vacuum creates an upward force to suck the particles off the brush and flapper. It takes significantly less force to take the particles off the flapper than the brush. This is due to the resistance created in why the particles are still stuck on the brush. A hypothetical number of particles of dirt are picked up within one vacuum housing area using the size of the vacuum housing, efficiency, and the dirt per area ratio. The subsystem comprises of the vacuum area of the Roomba including: the motor of the agitator and flapper, vacuum opening, and vacuum fan. Figure shows a cross-section of the Roomba from the patent. The housing opening of the Roomba holds the agitator and rubber flapper and provides an area for the particles and air to enter the vacuum. The Roomba has an agitator which sweeps up dirt and other particles from the floor. The agitator pushes some of to the rubber flapper and some of the particles stick on the brush. The rubber flapper throws the particles up for the vacuum to take. In addition, the vacuum can take particles off the brush. A flow diagram of this process can be seen in Figure 3. A motor controls how fast the agitator spins and creates the force with the carpet to draw the particles off the floor. Finally, the vacuum contains a fan to help create the suction for the vacuum. A motor controls the speed of the vacuum. Both of these motors consume current from the NiMH battery which controls the system. Agitator Flapper Figure. Cross-Section of Roomba Labeling the Agitator and Flapper 7

8 Tradeoffs exist throughout this optimization problem which makes it very interesting. First, an increased vacuum housing area will give more area to pick up dirt. However, it will create a larger agitator which will use more current to output the same amount of torque. In addition, a larger vacuum area will create less of a suction area by the fan due to Bernoulli s Equation. In addition, the sum of the current used by the two motors is limited by a constraint. It is interesting how the current will be distributed to each motor to satisfy the constraint..1. Mathematical Model Objective Function Figure 3. Flow Chart of Vacuum Subsystem The objective function is to maximize the amount of dirt that is picked up by the Roomba after going over the floor a specified number of times. In the current Roomba model it travels over a piece of floor four times before completion. Maximize ε T dirt (1) The total amount of dirt in a given area (dirt) is a product of a ratio of dirt per area by the area. The area is calculated by the length of the agitator by the amount the agitator touches the carpet. The distance of the radius of the agitator is displayed in Figure 3. This total amount of dirt is needed to calculate the objective function. dirt dirt r l ( Diam h) () A 8

9 Figure 4. Pictorial Explanation of Equation # The total efficiency is the equal to the sum of the efficiency multiplied by the available area left. For example if the efficiency is 75%, 100% of the dirt will be available to be swept up on the first try. During the second try to sweep the dirt, the vacuum will be 75% effective of sweeping the 5% remaining dirt. This will occur until the Roomba is programmed to stop which is a parameter. N 1 0 N ε T ε (1 - ε ) (3) The efficiency is calculated from the measuring the force of the vacuum and agitator. The force of the vacuum is measured using the pressure created by the fan. The vacuum fan creates a larger exiting air flow rate than on the inlet tube is. Due to the conversation of matter, a pressure difference is created known as Bernoulli s Equation as shown in Equation 4. Our control volume of our vacuum is shown in Figure 4. We assume that the pressure is on the outside of the fan is that of ambient pressure. In addition, we assume the height difference of approximately an inch is negligible. Finally, we assume the velocity of the air to be zero when the pressure hits it. P A ρ v A + ρ g h A P B ρ v B + ρ g h B (4) P 1 P ρ v B (5) Figure 5. Summary of Control Volume 9

10 The velocity of the fan is equal to the flow rate of the fan divided by the area of the fan. These two values are both parameters are shown in Equation 6. Flow V v B (6) AV The force of the vacuum on the agitator area is determined by dividing the pressure by the area that air may escape. F V P 1 l w (7) The force of the agitator is calculated determining the torque of the agitator which is used to calculate the total amount of current used by the motor. First, the resistance is calculated in Equation 8. The voltage, back-emf constant, torque constant, and speed constant are parameters for a specific motor. The torque of the motor is calculated as a function of the resistance in Equation 9. R A V κ ma ( κ ba ) (8) S A κ fa V A κ ma gear (9) R A Using the torque of the motor, a force the agitator outputs onto the carpet can be calculated. The agitator radius will vary with the size of the vacuum opening. The force of the agitator is calculated in Equation 10. It takes into account the force by the motor of the agitator and the kinetic friction in-between the brush and motor. The kinetic friction is proportional to the area and density of the agitator. F A 5 μ A ρ A k a a (10) 0. AA A A l w (11) R agit The efficiency of dirt picked up from the agitator controlled by the force of the agitator, length and the diameter of agitator. A parameter is set that calculates the amount of dirt that is pushed to the flapper as opposed with dirt stuck in the brush. It requires less of a force to remove dirt off the flapper than the brush as calculated through the constants. F A ε a (1) κ A ε b (1 perc flap κ VA ) ε a F V (13) 10

11 ( perc ε v flap κ ) ε F VF a V (14) ε ε b + ε v (15) Power Consumption To limit the speed and size of the motors the power consumed for each motor was calculated. The current of the agitator motor is calculated using the torque of the motor and torque constant. i A K ma A gear (16) The current of the fan motor is linear to the flow rate outputted. The fan characteristics supplied are the current used for the maximum flow rate. Thus a linear line can be interpolated for all flow rates used. i v Flow i Flow V v max (17) MAX The sum of the currents from the agitator and vacuum equals a total current used by this subsystem. Constraints i i + i (18) T A v Physical constraints l + w D R (19) Equation 16 is a geometric constraint which states that the length and width of the vacuum opening must not be larger than the diameter of the Roomba. i T i MAX (0) Equation 17 is a physical constraint which gives a maximum current the two motors can use. This constraint limits how big the motors can be which creates some design tradeoffs. Flow v Flow MAX (1) Equation 18 is a physical constraint which limits the flow rate of the vacuum fan. This maximum flow rate is motor is a characteristic of the maximum flow rate of the fan. 11

12 Practical constraints S A, FLOW V 0 () Equation 19 is a practical constraint which ensures the design variables of the speed of the agitator and flow rate vacuum fan are both positive. A negative speed or flow rate would not be desirable. ε 100% (3) Equation 0 is a practical constraint which ensures the efficiency of picking up dirt is less than 100%. A value greater than 100% is not a valid efficiency. Design variables and parameters Variables The list of design variables can be seen in Table. These variables are varied in the optimization model to help optimize the objective function. l w S A Flow V Design Variables Length of Opening Width of Opening Speed of Agitator Flow Rate of the Fan Table. Summary of Design Variables Figure 6. Summary of Bottom of Roomba 1

13 Length of Vacuum Opening l: The length of the vacuum opening can be found in Figure 4. This length will be perpendicular to a Roomba that is traveling forward. This design variable is dimensioned in meters (m). Width of Vacuum Opening w: The width of the vacuum opening can be found in Figure 4. The width is parallel to a Roomba that is traveling forward. This design variable is dimensioned in meters (m). Speed of Agitator ω A : The speed of the agitator is the speed in which the agitator of the Roomba travels while sweeping up the dirt. The design variable is dimensioned in rad/second (rad/s). Flow Rate of the Fan Flow V : The flow rate of a fan is a characteristic of a fan supplied by all suppliers. This design variable is measured in sec 3 m. Parameters These parameters are constants during a design optimization. This section will describe all of the parameters Number of Passes N: The number of passes of the Roomba is the number of times the Roomba travels over a spot before it decides it is clean. On the current Roomba design, the vacuum travels over every spot on the carpet four times on average. This parameter does not have a dimension. Ambient Pressure P : The ambient pressure is the pressure of the air outside of the vacuum. This parameter is measured in Pascal (Pa). Density of Air ρ: The density of the air is used in the measure of the effectiveness of kg the vacuum. This parameter is measured in kilograms by meters cubed ( 3 ). m Ratio of Dirt per Volume dirt r : The ratio of dirt per area is the number of dirt particles per a given area. This parameter is used to determine that amount of dirt that is picked up in the vacuum. The area in which the dirt occupies includes the length, width of the vacuum opening, and the diameter of the agitator. This parameter is measured in particles. 3 m Diameter of Roomba - D R : The diameter of the Roomba is the diameter of the complete vacuum. This is used in a constraint. The diameter of the Roomba is measured in meters (m). 13

14 Vacuum Area Covered by Agitator A A : This is the vacuum area covered by the agitator. This is used to measure how much air can escape as part of the vacuum. This parameter has no dimensions. Back-EMF Constant κ ba : The Back-EMF Constant of a motor is a characteristic which measures the Electromagnetic Force that is counteracting the torque of the motor. V s It is a constant and has dimensions of. rad Torque Constant κ ma : The Torque Constant of a motor is a characteristic which helps relate the torque of a motor to the inputted current. This parameter has dimensions of N m. Amp Rotational Friction of Motor κ fa : The rotational friction of the motor is a constant which gives on much friction is acting against the motor within it. The dimensions for this parameter are N*m-s. Voltage of the System V: This is the voltage supplied by the battery to the entire system. The current Roomba battery is a NiMH battery is fully charged at 18 volts [1]. It will be assumed we are using the model at 18 volts. Maximum Current Used i max : The total current used is the amount of current allotted to be used by the two motors. This parameter is measured in amps. Area of the Fan - A F : The area of the fan is the cross-sectional area of the fan across the blades. This parameter is in m. Maximum Flow Rate of Fan- Flow MAX : The maximum flow rate of the fan is the 3 m largest flow rate the fan can product. It is measured in. sec Vacuum to Agitator Efficiency Coefficient κ VA : The efficiency coefficient is a parameter which specifies the efficiency is in-between vacuum and the agitator. Agitator Efficiency Coefficient κ VA : The efficiency coefficient is a parameter which the efficiency is in-between agitator and the ground. Vacuum to Flapper Efficiency Coefficient κ VF : The efficiency coefficient is a parameter which specifies the efficiency is in-between vacuum and the flapper. Maximum Current for Fan - i VMAX : The maximum current for the fan is the current supplied at the maximum flow rate of the fan. It is measured in amps. 14

15 Agitator Ratio R agit : The agitator ratio is the ratio of the agitator size with respect to the vacuum opening. In the Roomba the agitator and flapper take up space in the vacuum opening. This parameter specifies the size of the agitator with respect to the vacuum opening. Percentage to Flapper Perc flap : The percentage to the flapper is the percentage of the dirt that the brush will push to the flapper. The actual percentage depends on the characteristics of the dirt pieces. Density of the Agitator ρ a : The density of the agitator is the density of the complete agitator including the brushes and post. Kinetic Friction Constant μ k : The kinetic friction constant is in-between the agitator brush and the carpet flooring. Dirt per Area dirt h : The parameter of dirt per height is the hypothetical amount of dirt particles per Gear Ratio on the Agitator Motor gear: The gear ratio on the agitator motor is the ratio in which the torque of the motor is amplified onto the agitator shaft. Summary of Model A summary of the subsystem optimization can be seen below: N 1 0 N Minimize - ε (1- ε ) dirt Where dirt dirt r l ( Diam A h) ε ε b + ε v (1 perc flap ) FA FV ε b κva κ A (1 perc flap ) FA FV ε b κ VA κ A FA ε a κ A A FA 0.5 l w R agit μ k Aa ρ a V A κ ma gear R A V κ ma R A ( κ ba ) S A κ fa F V P 1 l w 15

16 Flow v B A V V P 1 P ρ Flow AV V Subject To: Where: g 1 : l + w - D R 0 g : i + i i A v max 0 g 3 : Flow v - Flow MAX 0 g 4 : ε - 100% 0 g 5 : Flow v - Flow MAX 0 i i A v A K ma gear FlowV iv Flow MAX max Model Transformation The model took on transformations throughout its evolution. First, we planned a model before ever receiving a Roomba. In our original model we did not take into account the rubber flapper, it was believed that the system only contained an agitator. Thus, an algorithm was created to help take into account the rubber flapper. Also, receiving the Roomba allowed us to determine a gear box was used with the motor. Thus, we had to determine a parameter to represent the gear box. While using the Roomba it was discovered the bristle length was important is how much dirt was picked up. Thus, a three dimensional dirt density parameter was determined to take into account the bristle length instead of the previous two dimensional density parameter. The bounds of our design variables changed through our model transformation. When we assumed two of the constraints were active and there were two degrees of freedom in our problem, a three dimensional plot was performed with respect to two design variables. From these plots, it was seen that some the model was inaccurate for some of the design variable values. Thus, the bounds were reduced to take into account the values in which the model was feasible for..1.3 Model Analysis Examining the optimization problem can help determine how many degrees of freedom. First, four design variables exist from the model. The subsystem has no equality constraints on the design variables. Thus, the degrees of freedom of the problem are four. However, examining the inequality constraints tells us the maximum current and fan flow 16

17 rate have a large probability of being an active constraint. Thus, the degrees of freedom could be reduced to two. A monotonicity table of our problem can be found in Table 3. Monotonicity Table l w S A Flow V f U U + + g g - - g 3 - g 4 U U - - g 5 + Table 3: Monotonicity Table From the monotonicity analysis it can be seen by MP1 with respect to S A that the design variable S A will be bounded from above by g or g 4. It makes sense physically as the speed increases the current will increase and a constant is implemented on how large the current can grow. In addition, by MP1 with respect to Flow V that either g or g 3 will bounded by the flow rate from above. However, in most cases only one of these constraints, thus each one is conditionally active depending on the parameter values. From the monotonicity analysis it is unclear whether l and w will be increasing or decreasing with respect to the objective function and constraint g 4. However, it can be stated that the signs of the monotonicity table for l and w with respect to f and g 4 will be opposite. Also, the objective function is trying to increase the efficiency and constraint g 4 is putting an upper limit on the efficiency. Thus, there could be a condition on which g 4 is an active constraint. The constraint g 4 will be either active for all the variables or inactive for all the variables. Assuming the constraint will be inactive and l and w are gives us by MP1 with respect to l and w constraints g 1 could be active. Table 4 shows the results of these assumptions. From this model, it can be shown that g 1 will be active for the variables l and w by MP1 with respect to l and w. Monotonicity Table l w S A Flow V f g g - - g 3 - g g 5 + Table 4: Monotonicity Table assuming l and w are increasing in objective function 17

18 Scaling Scaling was an important part of our subsystem analysis. All of the design variables were scaled so their lower bounds were equal to zero and their upper bounds were equal to one. This allowed the algorithm to take equal steps for each of the design variable. In addition, the objective function is scaled to produce a larger gradient for the algorithm. Both scaling actions were very important is producing a robust model. Optimization Study From testing the model, it was evident that two constraints were always active. These two constraints were the maximum current limit (g ) and the geometric constraint limiting the size of the vacuum opening (g 1 ). Assuming these two constraints were active, two of the variables were solved using these equations as equalities. From this, we were able to plot the design space with respect to two variables at a time. The four plots are shown through Figure Figure 7. Design Space of Objective with respect to Length and Flow Rate of Exiting Fan 18

19 Figure 8. Design Space of Objective with respect to Width and Flow Rate of Exiting Fan Figure 9. Design Space of Objective with respect to Width and Speed of Agitator 19

20 Figure 10. Design Space of Objective with respect to Width and Speed of Agitator These figures accomplished two items. First, plotting the design space allowed me to recognize any irregularities in my modeling. I learned if I started the algorithm at points that violate a constraint, my model becomes inaccurate and the algorithm has trouble finding feasibility. This allowed me to choose the appropriate bounds for my design variables which my model was accurate for. In addition, plotting the design space showed that there was one global solution to the problem. Numerical Results The vacuum subsystem optimization was coded in MATLAB using the fmincon optimization. The variables were scaled from zero to one to help eliminate discrepancies in their magnitude. In addition, the standard convergence values of fmincon were used. First, a study was conducted to test for possible local minimas. This was conducted by starting the algorithm at different starting points and observing their behaviors. It must be noted that for some of the starting values the values for the flow rate of the fan or speed of the agitator went to the lower bound which is infeasible. It will be investigated for the reason of this occurrence. Table 5 shows a general result with all the parameters used. Table 6 analyzes the model from starting at three different initial points. All three of these starting points led to the same optimum. Using this information with the plots of the design space as shown in Figure 7-10 show there is a global solution to our problem. 0

21 Objective Function 18,015 particles/m Design Variables Lower Bound Upper Bound Starting Value l Length of Opening m w Width of Opening m S A Speed of Agitator rad/s Flow V Flow Rate of the Fan m 3 /s Constraints l + w D R i T i MAX Flow v Flow MAX S A, FLOW V 0 ε 100% Active? Yes Yes No No No Parameters Value Unit N Number of Passes 4 P Ambient Pressure Pa ρ Density of the Air 1.9 kg/m 3 dirt r Dirt to Area Ratio 5,000,000 particles/m Agit R Ratio of the Agitator to Flapper 50% D R Diameter of Roomba 0.18 m κ ba Back-EMF constant of Agitator Motor 8.4E-03 κ ma Torque constant of Agitator Motor 6.57E-0 κ fa Speed Constant of Agitator Motor 3.3E-04 V Voltage of the System 18 Volts i MAX Maximum current used 0.5 Amps A V Area of the Fan 0.03 m Flow MAX Maximum Flow Rate of Fan 0 m 3 /s κ Efficiency Coefficient 1 i VMAX Maximum Current for Fan 0. Amps μ k Kinetic Friction Constant 0.8 Perc flap Percentage to Flapper 50% gear Gear Ratio on the Agitator Motor 5 h Height of Agitator to Ground 0.0 m ε A Efficiency of Agitator 70 ε B Efficiency of Vacuum to Agitator 9 ε V Efficiency of Vacuum to Flapper 18 ρ a Density of the Agitator 0.5 kg/m 3 Table 5: Result of Subsystem Optimization with Parameters Trial #1 Trial # Trial #3 Design Variables Lower Bound Upper Bound Initial Point Final Point Initial Point Final Point Initial Point Final Pont l Length of Opening w Width of Opening S A Speed of Agitator Flow V Flow Rate of the Fan Objective: 18,00 Objective: 18,00 Objective: 18,00 Table 6: Subsystem Optimization to Investigate Local Minima In conclusion, the monotonicity analysis agrees with the numerical results. The monotonicity analysis stated that the packaging and current constraint might be active, which they are in the model. The KKT conditions could not be calculated as the 1

22 gradients of the objective function were unavailable to calculate. In addition, the global optimum occurs in the interior of the design variable s space but occurred where two inequalities are active. Parametric Study One of the most important parameters that dictate the optimization model is that the limit of current that can be consumed. This parameter decides the current to be used by the agitator motor and flow speed of the air exiting the fan. Thus, a parametric study was conducted by varying the maximum current that can be consumed the system. A table of the results can be seen in Table 7. It is very interesting that the vacuum opening length and width was almost constant for the different parameters. This might suggest that there is a golden ratio which is optimal for all the settings of the vacuum. In addition, the objective function increases quadraticaly while adjusting the maximum current. This result makes sense as the agitator and vacuum efficiency increases linearly, thus the sum of them would increase quadraticaly. The gear ratio was another parameter in which is easy to change in a design. Increasing the gear ratio increased the objective function which makes sense. This shows that you can continually increase the gear ratio to improve performance. A cost function could limit the gear ratio design as larger gears increase the cost. Maximum Current Design Variables units l Length of Opening m w Width of Opening m S A Speed of Agitator rad/sec Flow V Flow Rate of the Fan m 3 /s Objective Function particles/m,40 4,980 9,860 18,10 3,310 Gear Ratio Design Variables units l Length of Opening m w Width of Opening m S A Speed of Agitator rad/sec Flow V Flow Rate of the Fan m 3 /s Objective Function particles/m 4,980 8,70 1,830 Table 7. Parametric Study with respect to Maximum Current and Gear Ratio Discussion of Results The results of our optimization show insight into the practical design. The parametric study shows that the length and width of the opening are the same for all currents used. This is a satisfying result for the design team as it shows that they do not need to take into

23 account the amount of current being inputted into the system while deciding on the dimensions of the opening. If they were different, the designers could distribute a probabilistic function to each current consumed to determine which one is most important. In addition, the fact that the objective function increased quadratically with respect to increasing the maximum current makes sense as the objective function is linear to both the speed and flow rate of the fan. The model limits the solution as it does not take cost into account. Designing of consumer products are driven by low-cost manufacturing. However, it is unclear if the desired flow rate and speed of agitator is costly to replicate. Adding a cost function with respect to the fan and motor would be a smart way to add complexity to the model. Also, modeling the efficiency coefficients was difficult in the model. It was difficult determine the force necessary to pick up a very light weight particle that is stuck to a carpet fiber by friction. Thus, these efficiencies that were very rough estimates which make the objective function somewhat inaccurate. However, these efficiencies modeled the proportional properties very well as it was much easier for the vacuum to pick up the dirt off the rubber flapper than the agitator brush. 3

24 Subsystem : Driving wheels including a motor..1 Problem Statement As a part of the vacuuming robot, Roomba, the driving wheel with a motor is one of the key parts to accomplish its goal, cleaning up the house. Apart from a traditional vacuum machine, this Roomba requires a power train system to operate itself and this part leads to more complicated control system. Therefore, in this subsystem, it is necessary to make a good assumption to over come this complexity such as the Roomba s behavior using approximately 10 sensors, many of nonlinear parameters from a battery and a motor. First of all, the random movement of the Roomba might be necessarily determined with 3 different types of movement, a forward movement using wheels at the same time, a spiral movement using only one wheel especially when it is picking up more than usual amounts of dirt and a spinning movement using wheels to when turning its direction at the same position. For those 3 different cases, the power which a motor consume must be different. Second, it might be also reasonable to assume that there is no voltage drop for one battery life from a full charged through an empty state. Third, the velocity shape function is not linearly increasing with respect to time such that one proper function is necessary to develop for the evaluation of the movement. Based on these assumptions, the objective function of this subsystem is to minimize the energy consumption with one nominal path. The anticipated trade-offs are if we would like to maximize the speed of body which increase the area the Roomba travels, then the torque is decreasing and it reaches lower than the required torque for operation. Moreover, the radius of wheels must optimized between the size to maximize the speed of the Roomba and the one to minimize of torque applied by the ground... Mathematical Model Shape of discharge curve for typical batteries A typical discharge profile is shown below and it indicates that it is reasonable to assume that the voltage is constant during the battery discharging life. Voltage of the battery used for this model is 1 volts and the capacity of the battery is 3 Amp hour. 4

25 Voltage rate (%) Capacity Discharged Rate (%) Figure 11. Shape of discharge curve for a battery Principles of DC motors DC motor linear model derivation + V _ R m L V emf + _ motor J applied K f Figure 1. Typical DC motor model The first step in the derivation is to model the current through the motor. Using Kirchoff s current law, the linear equation can be written as di V Rmi L Vemf 0 (4) dt Using Newton s Law of Motion, the sum of the moments about the axis of revolution is M motor K f ω applied Jα (5) The torque in the motor is related to the current through the armature by motor K m i (6) and the back-emf voltage is related to the armature speed by 5

26 V emf K b ω (7) At steady state conditions, the acceleration of the motor and the change in motor current are zero, thus Rm i + K bω V (4)' K i ω 0 (5)' m K f Torque and speed of rotation The graph below shows a torque vs. speed curve of a typical DC motor. Torque is inversely proportional to the speed of the output shaft. In other words, there is a trade-off between how much torque a motor delivers, and how fast the output shaft spins. Motor characteristics are frequently given as two tow points on this graph. Motor Torque (Nm) Stall torque, s No load speed, ω n Figure 13. Torque vs. speed curve for a motor Rotational Speed - The stall torque, s, represents the point on the graph at which the torque is a maximum, but the shaft is not rotating. - The no load speed, ω n, is the maximum output speed of the motor when no torque is applied to the output shaft. From figure 13 above the relationship between the motor torque and the rotational speed can be derived as shown below (8) s ω + ωn s Using the equation (4)', (5)' and (6), the stall torque and the no load speed can be derived as shown below. 6

27 K m n V RmK f KbK ω (9) + m V s K m (30) R Torque and supply current m The torque and supply current relationship is also important characteristic of a DC motor. It is linear and is used to calculate the no load current and the current with the rotor stationary (start-up current). The graph for this relationship does not vary with the supply voltage of the motor. The end of the curve is extended in accordance with the torque and the start-up current. The gradient of this curve is called the torque constant of the motor. Motor Torque (Nm) Stall torque Current (no load) Starting Current i (A) Figure 14. Torque vs. current curve for a motor i 1 + i0 (31) K m Gearbox characteristics Gearboxes have been designed for optimum performance and for maximum life under normal operation conditions. Their main characteristic is the capacity to withstand maximum design torque with continuous duty. For the gearbox the selection criteria may be applied, thus R N required motor (3) N motor required 7

28 Some patterns of the Roomba Simplified nominal path of the body The movement of Roomba s body is regulated to simplify the pattern and it could be specified with 3 different types as shown below. Velocity (m/s) v 1 &v 31 v 3 v Case 1 Case Case 1 Case 3 d m d m time (sec) Case 1 Case 1 Case 1 Case 3 Case Dirt Area Figure 15. Simplification of the Roomba s movement Corner The First one is a straight movement of 1.5 meters with pattern (1). The next step is intensive cleaning mode because the Roomba detects dirt area using dirt sensor and it is classified as pattern (). After the intensive cleaning mode the Roomba is moving forward until it gets to the corner and it is trying to escape from the corner using both pattern (1) and pattern (3) together. 8

29 Case 1 : Forward movement using wheels with different speeds F F F r Figure 16. Free body diagram for forward movement From the figure 16 above, it is required to move at an expected constant velocity that the applied torque must be same amount with the force times the radius of a wheel. Therefore, w1 dv F1 M 1 x M 1 r dt r M 1 x motor w1 R r R M 1 x motor Case : Spiral movement using 1 wheel 1 (33) Fixed point I ω F Figure 17. Free body diagram for spiral movement Case illustrates how the Roomba behave to clean intensively for the dirty spot. To express this mechanism the following formula is used. 9

30 ) ( l x I R r R l x I r l l v l I r I l r l F motor motor w w w θ ω θ θ (34) Case 3 : Spinning movement at the same spot using wheels F F Fixed point ω 3 Figure 18. Free body diagram for spinning movement Figure 18 shows how the Roomba moves to turn its body at the same spot to escape from the corner and it is expressed as equation (35) ) ( l I R r R l I r l l v l I r I l r l F motor motor w w w θ θ θ ω θ θ (35) Velocity formula shape To modify the velocity change with respect to time, some assumptions are made. First of all, same velocity shape is used to express 3 different patterns and it is basically s-curve equation as shown in Figure

31 Figure 19. Velocity formula To express the velocity curve equation (36) is used. t vi v( t) (36) t + exp( a b t) where, t i t v d i i t + exp( a b t dv dt i 0 ) dt vi t + exp( a b t) t v (1 b exp( a b t)) i ( t + exp( a b t)) Objective function The objective function of this subsystem is to minimize the total efficiency of each pattern of the Roomba s movement. The total efficiency includes all 4 different types of behaviors through 1 battery life cleaning performance. t 1 Minimize f i1 ( t) dt + i ( t) dt + 5 i31( t) dt + 5 i3 ( t) dt (37) where, i i 0 s r t vi + i K m R ωn t + exp( a b t) t 0 t s + K m t 3 0 Constraints Physical constraints 1) Motor torque limited by stall torque ( s ) The torque generated by a motor is related to the acceleration, which is the differential of velocity, and it is not allowed to exceed the stall torque. 31

32 Max Max Max Max w1 s (38) w s (39) w31 s (40) w3 s (41) If the equation from (38) through (41) are rearranged, then dv Max dt dv Max dt v1 t v1 (1 b exp( a b t)) Max t t exp( a b t) ( t exp( a b t)) 1 dv Max dt dv Max dt v t v (1 b exp( a b t)) Max t t exp( a b t) ( t exp( a b t)) v1 t v1 (1 b exp( a b t)) Max t t exp( a b t) ( t exp( a b t)) r R M l r R I 31 v3 t v Max t t exp( a b t) (1 b exp( a b t)) ( t exp( a b t)) 1 r R M 1 l r R I s s s s (38)' (39)' (40)' (41)' Practical constraints 1) The radius of wheels The radius of wheels could not exceed a limited value because of the design constraints. If the radius is larger than the limit the Roomba will not be allowed to clean the floor underneath some kinds of furniture. At the same time, the radius must be larger than the lower limit, which is determined by the height of the body of the Roomba, and the values are used as 0. and 0.4 respectively. r lower lim it r r (4) upper lim it ) Gear ratio of gearbox The gearbox is usually used to reduce the speed, therefore, the required speed is less then motor speed and the gear ratio must be less than 1. Also the ratio is usually larger than small value and this is assumed based on the commercial production of the gearbox. Therefore, for this formula, R lower limit is assumed as and R upper limit as 1. R lower lim it R R (43) upper lim it 3) Limitation of velocity curvature at the initial point (t 0) 3

33 The velocity is increasing from zero to a certain amount, and it is impossible physically, at that moment, to increase rapidly. Therefore, it is assumed that the gradient at the initial point does not exceed 0.1. Thus, dv( t) v1 dt exp( ) t 0 a dv( t) v dt exp( ) t 0 a dv( t) v3 dt exp( t 0 a ) (44) (45) (46) To reduce the cancellation error when large number is located in the denominator, equations from (44) through (46) are manipulated as following and those equations are still valid because the term exp(a) will not be equal to or less than zero. v exp( a) 0 (44) v 0.1 exp( a) 0 (45) v 0.1 exp( a) 0 (46) 3 Design variables and parameters Design variables a b r R Velocity function variable Velocity function variable Gear ratio of gearbox Radius of a wheel (m) Table 8. Summary of Design Variables Design parameters V Voltage of a battery (V) R m Terminal resistance in a motor (Ω) L Rotor inductance in a motor (H) Back-EMF voltage (V) V emf J Rotor inertia (kg m s ) Torque Constant (N m/a) K m K b K f s ω n i i i o Back-EMF constant (V s/rad) Rotational friction constant (N m sec) Stall torque (N m) No-load Angular velocity (rad/sec) Start-up current (A) No-load current (A) 33

34 d i v i M 1 Distance of body movement (m) Desired velocity (m/s) Mass of the Roomba s body (kg) I i Moment of inertia (N m sec ) Table 9. Summary of Parameters Summary model Minimize f t 1 0 where, i t i1 ( t) dt + i ( t) dt + 5 i31( t) dt + 5 i3 ( t) dt i 0 s r t vi + i K m R ωn t + exp( a b t) t t 3 0 s + K m subject to g1 : Max Model analysis w s g : Max 0 w s g3 : Max 31 0 w s g4 : Max 3 0 w s g5 : r lim r 0 lower it r upper g 6 : r lim 0 g7 : R lim R 0 lower it R upper g 8 : R lim 0 g 9 : v1 0.1 exp( a) 0 g 10 : v 0.1 exp( a) 0 g 11 : v3 0.1 exp( a) 0 it it Before starting the optimization study, it might be helpful to check whether this subsystem could be simplified by some algebraic manipulation. However, unfortunately, there exist some difficulties to perform the monotonicity analysis analytically because it is not possible to have the explicit mathematical model as seen from the summary model above, for instance, the objective function and the constraints from g1 through g4. Therefore, numerical method might be the only one alternative way to check the monotonicity properties for this subsystem model. All parameter values related to the motor for the subsystem analysis are from the values for the motor used in the Roomba vacuum and other parameters are based on the measurement of the Roomba. 34

35 Monotonicity analysis First of all, the objective function is evaluated with respect to the velocity shape function variable a which determines the gradient of the s-shape function. The other variable values are fixed. Figure 0 shows how the design variable a affects the objective function Objective Function (Amp*sec) Velocity shape function variable 'a' Figure 0. Objective function evaluation with respect to the variable a From the Figure 0, it tends to be a monotonic in the boundary region, but it could not be ensured because there are several flat areas. Also the difference between the values of objective function is not sufficiently large, which means that this function is fairly flat. The similar evaluations for the objective function are executed to check the monotonicity and the results are attached on Figure 1 through -.- below Objective Function (Amp*sec) Velocity shape fuction variable 'b' Figure 1. Objective function evaluation with respect to the variable b Figure 1 shows that the objective function is decreasing exponentially with respect to the variable b increasing and it is monotonic. 35

36 Objective Function (Amp*sec) Radius of wheel 'r' (m) Figure. Objective function evaluation with respect to the variable r From the examination of the objective function with respect to the increasing variable r, the objective function tends to decrease linearly and if there is no active inequality constraints to the effect of this variable on the objective function, then the lower boundary inequality constraint of the variable r might be assumed to be active. However, the difference between the largest and lowest values of the objective function is not much large. Figure 3 below shows that the increasing variable R causes the objective function to increase rapidly for the small R and the rest of region has approximately same objective values Objective Function (Amp*sec) Gearbox ratio 'R' Figure 3. Objective function evaluation with respect to the variable R From the monotonicity check, the objective function has clearly monotonic maneuver with respect to the design variable b, r and R respectively except the one a. As a result, the evaluation of objective function is carried out with respect to a and b together to test the subsystem model and the result are plotted in Figure 4 below. Most of the design variable space seems to be flat and that area is magnified. 36

37 ..4 Optimization Study Figure 4. Objective function evaluation with respect to a Algorithm used and scaling To execute the optimization study, the fmincon function equipped in the MATLAB is used and the algorithm of fmincon is based on the SQP (Sequential Quadratic Programming) approach. The SQP algorithm is very sensitive to the scaling such that initially all design variables are scaled from zero to one and the optimum variables are un-scaled when the iteration of algorithm terminates. The fmincon function tends to provide many different objective function values from different starting points and sometimes fail to find out the feasible optimum values because the subsystem model is not robust. From the previous model analysis, it might be thought that the computed values for the objective function from most of the design space are too close to apply the SQP algorithm, which is based on the gradient method. For the system robustness improvement to have only one global optimum from the feasible domain, the scaling of the objective function is also required. Therefore, the objective function is scaled up gradually and it could be observed that the subsystem 37

38 model gets more robust and the SQP algorithm tend to converge on the optimum value. After one optimum is found, of course, the value must be scaled down to have an exact answer. One of the results using an arbitrary starting points are distributed in Table -.- below. Numerical results Objective Function Value Unit f Electricity consumption 4.1 (Amp sec) Design Variables Lower Upper Starting bound bound Point Value Unit a Velocity function variable b Velocity function variable r Radius of wheels (m) R Gear ratio of gear Constraints Max w1 s Max w s Max w31 s Max w3 s r lower limit - r 0 r r upper limit 0 R lower limit R 0 R R upper limit 0 v 1 /exp(a) 0.1 < 0 v /exp(a) 0.1 < 0 v 3 /exp(a) 0.1 < 0 Activity Active Active Active Active Active Active Parameters M 1 Mass of body. kg I Moment of inertia with case N m sec I 3 Moment of inertia with case N m sec V Voltage of a battery 1 V R m Terminal resistance in a motor Ω L Rotor inductance in a motor 1.1e-3 H V emf Back-EMF voltage 8. V J Rotor inertia.3e-6 kg m s K m Torque Constant N m/a K b Back-EMF constant 4.1e- V s/rad K f Rotational friction constant 0.4e-5 N m sec s Stall torque 10.6e-3 N m ω n No-load Angular velocity rad/sec i i Start-up current 0.41 A i o No-load current 0.05 A d 1 Distance of movement with pattern (1) 1.5 m d Distance of movement with pattern () 8.64 m d 31 Distance of movement with pattern (3-1) 0.03 m d 3 Distance of movement with pattern (3-) m v 1 Desired velocity for pattern (1) 1 m/s v Desired velocity for pattern () 0.7 m/s v 3 Desired velocity for pattern (3) 1 m/s r limit Limit value of wheels 0.04 m Table 10. Result of Driving Wheels subsystem 38

39 From the Table 10, it is determined that some of the inequality constraints (g1, g3, g6, g7, g9 and g11) are active. Two active constraints, g6 and g7, are expected from the previous model analysis using monotonicity as mentioned before. For the given parameters value, two sets of constraints, {g1, g3} and {g9, g11} have exactly same condition because the desired velocity for pattern (1) and pattern (3) are same, which are essentially estimated by measuring the real Roomba s movement....5 Parametric Study Most of parameters related to the motor are determined together by a motor used If those parameters values are desirable for parametric study, it seems to be much more complicated and also it might not be physically meaningful. The parameters used for the parametric study here are the total weight, M 1 and the desired velocity for pattern (1), v 1. Table 11 shows the relationship between the weight of the Roomba and objective function. M a b r R f Table 11. Parametric study with respect to v 1 As expected intuitively, the increasing weight increases the current consumption, which means that more weight needs more energy to complete the given task. Therefore, it might be obviously tried to reduce the weight of the body from the other point of view. Table 1 shows the relationship between the desired velocity of the Roomba to operate and the objective function optimum. v a b r R f Table 1. Parametric study with respect to M1 From the result, it can be seen how the different desirable velocities affect the objective function of the subsystem. When the desirable velocity increases, the current consumption tends to be decreasing. However, considering the mechanism of the whole system, it might be noticed that there exist a physical constraint because the increasing velocity of the Roomba s body will makes the vacuuming efficiency worse. 39

40 ..6 Discussion of Results The subsystem optimization to achieve the objective is completed and the mathematical model of this subsystem is sufficiently robust to have one global optimum after several trial and error steps. To make the optimization successful, first of all, the model construction is carefully executed and many useful options must be considered, such as scaling of boundary and objective function. The result of the subsystem optimization will be used to adjust the Roomba s control mechanism to minimize the current assumption for the nominal task. The control system might make the Roomba s velocity profile based on the one determined by the optimum values using simple electric devices such as an amplifier, delay and so on. The optimum value of the wheel radius meets the physical constraint. If the allowable wheel radius is larger than a given value, then it might decrease objective function. However, unfortunately, if the Roomba has a bigger wheel, then it might lose its advantage to clean under some furniture. And the optimization results shows that the smaller gearbox ratio will decrease the current consumption and it also meets the allowed manufacturing limit and the cost of production. In general, this subsystem model is simplified to construct the mathematical model and there might exist several other ways to do it. One of the ways is that some important design parameters are changed to deign variables to optimize the system and this might be performed later on. 40

41 3. System Integration The objective of the system integration was to maximize the amount of dirt picked up before a certain amount of power is consumed. Originally, the amount of current consumed was the battery maximum. However, it was decided a user did not want the Roomba working for a very long time, so the redefined objective s goal is to maximize the dirt picked up in a short amount of battery use. It is assumed the same driving interval used in that subsystem is repeated until the current constraint is active. This accurately models a Roomba as it travels throughout the room finding dirt and avoiding obstacles. The system integration gave us several tradeoffs within each subsystem. The major tradeoff is how the current will be distributed to each subsystem. It will be interesting to tell if it better to have a high efficient vacuum that travels a short distance or a low efficiency that travels a longer distance. Integrating these two subsystems will help clear up this tradeoff. Second, if the suction power and speed of the vacuum is increased the battery life would decrease. A balance between the housing, motors, and battery size will be determined through the system optimization. All-in-One Approach In the All-In-One approach, each subsystem is combined together to create a system model. Each subsystem is integrated into the system with the addition of one design variable for a total of nine design variables. The new design variable is the number of times the driving interval is run. This variable is used to determine the total time the battery runs and the total distance traveled. Figure 5 shows what values are passed on from each subsystem to help solve the system problem. The vacuum subsystem is calculates the vacuum current but is multiplied by the time per interval to determine the amount of amp-hr current is consumed. The calculations of the equations can be found in equation (46). Figure 5: Summary of AiO System Integration 41

42 New AiO Design Variable Number of Intervals N int : The number of intervals is the number of times the driving cycle executed in the driving subsystem. It will execute these intervals until the battery dies. New AiO Constraint i SSV + i i SYSMAX (46) SSD Equation 46 is a physical constraint which gives a maximum current the two subsystems can use. This constraint limits how much dirt the vacuum can pick up before the battery dies. Objective Value Calculation The objective value is calculated by multiplying the distance traveled per interval by the number of intervals by the dirt picked up per meter. The original system design problem was having difficulty converging to an answer. We believe the large amount of design variables used in the AiO approach is the reason. From the subsystem optimization, a golden ratio of the length and width of agitator opening was determined. These values were the same for all of the optimization results at different parametric values. It was decided that the length and width of the agitator were to be parameters instead of design variables. The results from the AiO approach are presented in Table 13. Interdisciplinary Feasible (IDF) Algorithm Decomposition of the problem was used to attempt to help reduce the complexity of the problem and to converge quicker. It was determined that the current acted as a coupling in-between the subsystems, as the current of one subsystem is increased the current allocated to the other subsystem must be decreased. Thus an IDF formulation is developed with the two subsystems where two new design variables are added. With the IDF formulation, new two new equality constraints are added which state the current design variables must be equal to the analysis values coming from the subsystems. These design variables are essentially dummy variables to help with the decomposition. This decomposition could be practical in an organizational setting. 4

43 Figure 6. Summary of IDF Decomposition The IDF formulation uses the new design variables and constraints of the AiO problem, plus the addition of the following variables and constraints: New IDF Design Variables Driving Subsystem Current Allocated IA DSYS : The current that is allocated to the driving cycle subsystem. Vacuum Subsystem Current Allocated IA VSYS : The current that is allocated to the vacuum cycle subsystem. New IDF Equality Constraints IA VSYS i SSV (47) IA DSYS i SSD (48) The IDF objective function is calculated the same way as the AiO. Results and System Integration Discussion Design Variables Lower Bound Upper Bound Starting Value All-In-One Appraoch IDF Algorithm Units S A Speed of Agitator rad/s Flow V Flow Rate of the Fan m 3 /s a Velocity function variable b Velocity function variable r Radius of wheels m R Gear ratio of gear N int Number of Intervals IA DSYS Driving Subsystem Current Allocated amp-hr IA VSYS Vacuum Subsystem Current Allocated amp-hr Objective Function 1.89 E+8.01 E+8 Particles 43