Calculations different driving situations The worst case scenario the outdoor robot platform can be in is assumed while making calculations. The gear ratio between the motor and axle and the wheel axle is 1:1 unless otherwise specified. Calculation maximum speed outdoor robot platform: The maximum speed of the outdoor robot platform can be calculated with the formula below. 60 The motor makes 122 revolutions per minute at rated load 1 (Source: 1). 122 The circumference of the wheel can be calculated by multiplying the wheel diameter with pi 2 (Source: 2). 0,254 0,798 Then all the data is known to calculate the maximum speed of the outdoor robot platform. 122 0,7981,62 / 60 Calculation stall-torque motor: The motor delivers 1,77 Nm torque at 122 RPM. The no-load speed of the motor is 142 RPM. The motor will not deliver any torque at no-load speed. Because the relationship between torque and speed is linear, the stall torque can be calculated. The stall torque is the torque that the motor delivers when it is starting to turn and is also the maximum torque. )*-(*'. 142 $ %&'(( $ )*+,)'( 1,77 12,567 0 )*-(*'. / )*+,)'( 142/122 Figure 1 shows the torque speed characteristic of the motor. Figure 1: Torque-Speed characteristic 1 Datasheet motor 2 Datasheet wheel
Calculating acceleration outdoor robot platform: Figure 2: Free body diagram outdoor robot platform on flat ground The maximum mass of the outdoor robot platform and the total load it can carry is 35 kg. 35 23 The sum of forces is equal to the mass multiplied with the acceleration of the mass (Figure 2). The vertical forces are not used in this calculation because they do not contribute to the acceleration in horizontal direction. 45 6 7 6 The sum of forces exists out of a forward force that the wheels exert on the ground (F v ) and a friction force that is caused by friction between the wheels and the ground (F w ). 5 8 /5 9 7 6 The forward force is calculated by dividing the stall-torque of the motor which is 12,567 Nm by the radius of the wheel that is driven by the motor. The force per wheel is calculated in the formula below. $: <0= 5 8 12,567 98,950 > 7; 0,127 <= The friction force on the wheels is calculated by multiplying a factor with the weight that each wheel carries in the calculation below 3 (Source: 3). Other friction influences are neglected. 5 9 ~5% 5 A9B*C*& 35 9,81 0,054.29 0 > 4 Fill in the basic formula with the missing data. 98,95 /4,29 35 4 7 6 Then then formula can be defined to calculate the maximum acceleration (a x ). 98,95 /4,29 7 6 10,82 / E 8,75 This drive away acceleration is only present for a fraction of a second because the torque will lower when the speed increases to the nominal torque of 1,77 Nm. 3 Datasheet friction force percentages of different rubber wheels on the ground
$: [0] 5 8 1,77 = 13,9370 > h 7; 0,127 [] 13,937 4,29= 35 4 7 6 7 6 = 13,937 4,29 = 1,1 / E 8,75 The acceleration at nominal torque is 1,1 m/s². This means that the acceleration will lower 10,82 m/s² to 1,1 m/s² during the acceleration from stand still to maximum speed. The average speed during that period is FG,HE-F,F =4,86 /². The maximum speed of the outdoor E robot platform is 1,62 m/s which will be reached in F,JE =0,33 seconds. K,HJ It could occur that the outdoor robot platform is going to be in a situation where two wheels do not touch the ground anymore when driving over uneven ground. Only two motors can cause movement of the outdoor robot platform in this case. Calculations will determine the maximum acceleration from stand still and while driving in this situation. 5 8 5 9 = 7 6 5 9 =~5% 5 A9B*C*& = 35 9,81 0,05=8,58 0 > h 2 98,95 8,58= 35 2 7 6 7 6 = 98,95 8,58 = 5,16 / E 17,5 The acceleration of the outdoor robot platform will decrease when the outdoor robot platform starts to move until it reaches the constant acceleration calculated below. 13,937 8,58= 35 2 7 6 7 6 = 13,937 8,58 = 0,306 / E 17,5 The average acceleration during this period is L,FJ-G,MGJ = 2,427 m/s² and the maximum speed of the E outdoor robot platform will be reached in F,JE = 0,66 seconds. E,KEN The force that a wheel exerts on the ground is the same in every situation. But each wheel will have to carry more weight when using fewer wheels to drive with. When a wheel needs to carry more weight it results in a higher friction force between the wheel and the ground. This means that the force to create acceleration is lowered when a wheel needs to carry more weight. Thereby the acceleration is lower which increases the time that the outdoor robot platform needs to reach maximum speed.
Calculation acceleration outdoor robot platform on a slope: The maximum acceleration of the outdoor robot platform decreases when the outdoor robot platform drives up a slope because the force of gravity must be overcome. The force, displayed as 5 A96 in Figure 3, is a multiplication of the gravity force of the outdoor robot platform with the angle of the slope the outdoor robot platform is driving up. Figure 3: Free body diagram outdoor robot platform on ground with slope The mass and the load of the outdoor robot platform are also 35 kilograms in this situation. When the reference of the x-axis is parallel to the slope the outdoor robot platform is driving up, the following equation can be used. 45 6 = 7 6 The sum of the forces exists of a force forward that the wheels exert on the ground (5 8 ), a friction force of the wheels on the ground (5 9 ) and e gravity force in the x-direction by the outdoor robot platform (5 A96 ). 5 8 /5 9 /5 A96 7 6 5 8 and 5 9 are the same as in the situation where the outdoor robot platform is driving over a flat underground. The gravity force of the platform in x-direction (5 A96 ) can be calculated by multiplying the sine of the angle that the slope has with the gravity force 5 A9. 5 A96 sinr30s 9,81 35171,675 0 Fill in all the calculated values in the basic equation. 98,95/4,29/ 171.675 35 4 4 7 6 Then the maximum acceleration can be calculated when the outdoor robot platform starts to drive. 7 6 98,95/4,29/42,92 5,91 / E 8.75
The outdoor robot platform will start to move up the slope when the angle of the slope is 30. In practice the motors cannot get up to nominal speed but they will deliver more torque. This causes the motors to overload. The amount of overload is dependent on the speed that the outdoor robot platform can achieve and the angle of the slope. The maximum angle that can be driven up to with the ordered motors that deliver 12,567 Nm stall-torques can be calculated to set a limit to the angle that a slope can be. The free body diagram of Figure 3 will be used for this calculation with the same data. 45 6 = 7 6 5 8 5 9 5 A96 =0 5 8 5 9 =sinrts 3 The friction and forward force that the wheels experience will stays equal. Then the formula can be circumscribed and filled in to calculate the maximum angle. θ = sin -F R 5 8 5 9 S= sin-f98,95 4 4,29 4 =90 3 35 9,81 The outdoor robot platform will be able to drive up any slope. But the motors will be overloaded when driving up certain slopes. Hereby applies that the overload of the motors reduces when the angle of the slope that the outdoor robot platform is driving up decreases. If the slope reaches a certain angle the motors won t be overloaded anymore but they will be loaded with their nominal load. This angle will be calculated with the same principal. Force 5 8 changes to the nominal motor torque which is 1,77 Nm. Because the motor torque changes, the forward force per wheel will change. $: [0] 5 8 = =1,77 = 13,9370 > h 7; h 0,127 [] Then the angle for nominal load can be calculated for the outdoor robot platform. θ = sin -F R 5 8 5 9 S= sin-f13,937 4 4,29 4 = 6,45 3 35 9,81 By the above calculations can be concluded that the motors are not overloaded with slopes with an angle less than 6,45. The motors will be overloaded when the angle is larger than 6,45 and will be loaded heavier if the angle increases. It is not recommended to drive up slopes that heavily overload the motors because it reduces the lifetime of the motors and can even make the motors burn out.
When the outdoor robot platform is in a situation were only two wheels touch the ground on a slope, the maximum angle of the slope that can be driven up to changes. At nominal load the following changes can be noticed. $: [0] 5 8 = =1,77 = 13,9370 > h 7; h 0,127 [] 5 9 =~5% 5 A9B*C*& = 35 9,81 0,05=8,58 0 > h 2 θ = sin -F R 5 8 5 9 S= sin-f13,937 2 8,58 2 = 1,78 3 35 9,81 The following changes can be noticed at stall-torque. $: 12,567 [0] 5 8 = = = 46,30 > h 7; 0,127 [] 5 9 =~5% 5 A9B*C*& = 35 9,81 0,05=8,58 0 > h 2 θ = sin -F R 5 8 5 9 S= sin-f46,3 2 8,58 2 = 31,76 3 35 9,81 Only the wheels that make contact with the ground can exert force on the ground to move the outdoor robot platform. But when only two wheels touch the ground they will exert less total force to move forward and the friction per wheel increases. That causes the lowering of the maximum slope that the outdoor robot platform can drive up to under nominal load and it also overloads the motors faster.
Calculation for driving over an obstacle: The outdoor robot platform can come across objects on its way that it needs to drive over. The worst case scenario is shown in Figure 4. Hereby the outdoor robot platform is standing still against the object with one wheel. Figure 4: Free body diagram for driving over an object When the outdoor robot platform wants to drive over an object from standstill, there are no forces in x- direction if the object is large. This means that there has to be movement in the y-direction first before there can be movement in the x-direction. The movement in y-direction will be made by the motor(s) that drive the wheel(s) that are standing against an object. The other wheels will start to help the outdoor robot platform to move short after the movement in y-direction is made. 45 W = 7 W 1 2 5 ) X5 8W /5 A9 7 W 1 2 5 ) X5 8 sin RYS/ 3 7 W For this example, it is assumed that angle x is 45. Furthermore, there will be a friction of 4,29 newton per wheel. The static friction coefficient of the rubber wheel on the concrete obstacle is 1 4 (source: 4). 5 ) 5 9 4,29 4,29 0 Z % 1 2,15X98,95 sin R45S/35 9,8135 7 W 2,15X69.97/343,3535 7 W /271,2435 7 W 7 W /271,24 /7.75 /² 35 In this situation the outdoor robot platform will not be able to drive over the object and will remain standing against the object. 4 Static and dynamic friction coefficients on different materials
Bibliography 1. robots, Superdroid. Item details. Superdroid robots. [Online] [Cited: 02 06, 2013.] http://www.superdroidrobots.com/shop/item.aspx?itemid=849. 2. ATR Wheel and Shaft Set Pair 8mm bore - 10 inch pneumatic. Superdroid. [Online] [Cited: 02 06, 2013.] http://www.superdroidrobots.com/shop/item.aspx?itemid=1131. 3. Wicke. Productinformatie. Interwiel. [Online] [Cited: 02 06, 2013.] http://www.interwiel.nl/webroot/storenl/shops/61440372/mediagallery/pdf_downloads/productinf ormatie.pdf. 4. Wikipedia. Wrijvingscoëfficiënt. Wikipedia. [Online] 02 03, 2013. [Cited: 02 20, 2013.] http://nl.wikipedia.org/wiki/wrijvingsco%c3%abffici%c3%abnt.