# Relationship between motor torque and acceleration

I am working on designing and building a small (1 1/2 lbs), 2-wheeled, differential drive Arduino-controlled autonomous robot. I have most of the electronics figured out, but I am having trouble understanding how much torque the motors will actually need to move the robot. I am trying to use the calculations shown here and the related calculator tool to determine what speed and torque I will need. I will be using wheels 32mm in diameter and one of Pololu's High-Power Micro Metal Gearmotors. I performed the calculations for a robot weight of 2 lbs to be safe and found that the 50:1 HP Micro Metal Gearmotors (625 RPM, 15 oz-in) should theoretically work fine, moving the robot at 3.43 ft/s with an acceleration of around 29 ft/s^2 up a 5-degree incline.

However, I have not found an explanation for several things that I think would be very important to know when choosing drive motors. When the robot is not moving and the motors are turned on at full power, they should need to deliver their full stall torque. Based on the calculations, it seems that any amount of torque can get the robot moving, but the more torque, the faster the robot's acceleration. Is this true? Also, if the power source cannot supply the full stall current of the motors, will the robot not be able to start moving? In my case, I am powering the robot through a 7.2V (6S) 2200mAh NiMH battery pack that can provide around 2.6A continuously, and when it does that the voltage drops to less than 1V. Will this be able to power my motors? Once the robot reaches full speed and is no longer accelerating, theoretically the motors will not be providing any torque, but I do not think this is the case. Is it, and if so, how will I know how much torque they will be providing? Will the motors I chose have enough torque to move my robot?

• You are missing some key information about how DC motors work. – hauptmech Mar 12 '16 at 6:06
• One more thing, that I forgot to add: Pololu HP motors are highly stressed - they deliver high power from really small size. If you want to drive your robot for many many hours without risk of motors breking up, you may want to consider switching to a bigger motor with similar parameters. – mactro Mar 15 '16 at 9:00

## 2 Answers

1. Based on the calculations, it seems that any amount of torque can get the robot moving

That is true only for a theoretical robot moving in vacuum without any friction in its mechanisms. Normally if you limit a current too much, you won't even get the motor moving without any load, not to mention a whole robot.

1. the more torque, the faster the robot's acceleration

Yes, the acceleration is directly proportional to the force you apply $$a=\frac{F}{m}$$, which in turn is directly proportianl to the torque $$F=\frac{T}{R}$$ Both mass and wheel radius are constant, so when you increase torque, the acceleration increses also.

1. if the power source cannot supply the full stall current of the motors, will the robot not be able to start moving

The torque is proportional to the current you supply, so in the worst case scenario (depleated, undersized battery ant too weak motor) this may happen. However, for your robot and motors you should be fine. Your robot will just accelerate more slowly.

1. Once the robot reaches full speed and is no longer accelerating, theoretically the motors will not be providing any torque, but I do not think this is the case. Is it, and if so, how will I know how much torque they will be providing?

Again, you forget about the friction. Your robot will stop accelerating when friction will be equal to the torque. For Pololu motors this happens usually at 70-90% of motors no load speed, depending on robot's weight and mechanics.

Generally, in my calculations I usually choose 50% safety margin, and I have never had any trouble. You have some 30%, but that should be fine too.

1) but the more torque, the faster the robot's acceleration: YES $$F=\frac{Torque}{wheelradius}$$ and robot's acceleration is $$a=\frac{F}{m}$$ so, from that equation you can see that linear acceleration of the robot increases with the torque

2)Once the robot reaches full speed and is no longer accelerating, theoretically the motors will not be providing any torque:NO

According to the equation: $$T=\frac{P}{w}$$ that appears in one of the tutorials that you sent us, it is not possible to have 0 torque except that when the power=0. When you are at top speed, angular speed is high and torque is just enough to overcome the different resistances to the motion (friction, air resistance, whatever). It represents a very small value, which makes that the acceleration is close to 0, according to my first and second formula.