Sizing Electric Motors for Mobile Robotics

Sizing Electric Motors for  Mobile Robotics

May 21, 2006

The Basics

May 21, 2006

Unit Conversions 2π 

rad  sec

1Watt 

=

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=

1

rev sec

1Volt 

⋅

1Watt 

 Ampere

=

 N  m

=

1

1Volt 

⋅

sec

Coulomb sec

Basics The FORCE applied by a wheel is always tangent to the wheel.

Force is measured in units of weight w eight (lb, oz, N) May 21, 2006

Basics The required TORQUE to move a mobile robot is the force times the radius of the wheel.

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Torque is measured in units of weight w eight x length (lb·ft, oz ·in, N·m)

Procedure for Sizing DC Motors

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Information Needed • Estimated Weight • Number of wheels and motors • Maximum incline • Desired maximum velocity at worst case • Push/Pull forces

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Procedure • Step One: Determine total applied force at worst case

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Friction • Static Friction  – Used to determine traction failure

• Rolling Friction  – Used to determine motor requirements

• Kinetic Friction

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Rolling Friction  F  R

= µ  ⋅ N   R

Ro lling friction ∀ µR Is the coefficient of Rolling  – Using the coefficient of Static friction ( µS) will typically be to high

• To determine µR:  – Roll a wheel at a initial velocity, v elocity, v, and measure the time, t, in which it takes to v stop

 µ  R

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=

t   g  ⋅

Rolling Friction • Some typical values for µR  – Steel on steel: 0.001  – Rubber on pavement: 0.015

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Other Forces • Gravity  F  I 

= W  ⋅ sin θ 

• External θ

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Total Force • Calculate worst case  – Up hill with rolling friction  F  W  ( µ R cos θ 

= ⋅

⋅

+ sin θ )

 – Up hill with rolling friction, pushing  F  W  ( µ  R cos θ  sin θ )  – Level ground with rolling friction  F  µ  R W 

= ⋅ =

⋅

+

+ F 

 EX 

⋅

 – Level ground with rolling friction, pushing  F  May 21, 2006

= µ  ⋅W  + F   R

 EX 

Other Cases • Tracks  – Set µr =0  – Use a spring scale to determine the force required to pull the chassis in neutral and add that to the t he worst case force

• Gear Trains  – Bulky gear trains may significantly affect the outcome  – If this is a concern, it may be best to test in the same way as tracks May 21, 2006

Procedure • Step One: Determine total applied force at worst case • Step Two: Calculate power requirement

May 21, 2006

Power Requirement • Determine velocity, v, requirement under maximum load (worst case force) • Using the worst case force and velocity, calculate the power requirement  P 

=  F  ⋅ v

• This is the total power, divide by the number of motors if more than one motor is used RULE OF THUMB: 3 TIMES MARGIN May 21, 2006

Procedure • Step One: Determine total applied force at worst case • Step Two: Calculate power requirement • Step Three: Calculate torque and speed requirement

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Speed/Torque Requirements • Using the velocity requirement, v, and the radius of the wheel, r 

ω  =

v

r 

Speed requirement is in rad/sec

• Using the speed from above and the power per motor  T  May 21, 2006

=

 P 

ω 

Procedure • Step One: Determine total applied force at worst case • Step Two: Calculate power requirement • Step Three: Calculate torque and speed requirement • Step Four: Find a motor that meets these requirements May 21, 2006

Spec Sheet

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Spec Sheet

May 21, 2006

Procedure • Step One: Determine total applied force at worst case • Step Two: Calculate power requirement • Step Three: Calculate torque and speed requirement • Step Four: Find a motor that meets these requirements • Step Five: Plot motor characteristics May 21, 2006

Torque vs. Speed Curve T 

= T  −  PK 

T  PK  S  NL

• Where T = Torque • TPK = Stall Torque • SNL = No Load Speed ∀ ω = Speed May 21, 2006

⋅ ω 

Torque vs. Speed Curve Torque vs. Speed 7.00E-02

From this plot, maximum speed can be determined for a given load.

6.00E-02

5.00E-02

  m 4.00E-02    N  ,   e   u   q   r   o 3.00E-02    T

2.00E-02

1.00E-02

0.00E+00 0

1000

2000

3000

4000

Speed, rpm

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5000

6000

7000

8000

Power  T 

= T  −  PK 

T  PK  S  NL

⋅ ω 

ω  = (T  PK  − T )

 P   P (ω )

 P (T ) May 21, 2006

=− =−

= T  ⋅ ω 

T  PK  S  NL S  NL T  PK 

⋅ ω  + T  ⋅ ω  2

PK 

⋅ T  + S  ⋅ T  2

 NL

S  NL T  PK 

Power  Power vs. Speed 1.20E+01

1.00E+01

8.00E+00

  s    t    t   a   w  , 6.00E+00   r   e   w   o    P

 P (ω )

4.00E+00

2.00E+00

=−

T  PK  S  NL

⋅ ω  + T  ⋅ ω  2

PK 

0.00E+00 0

1000

2000

3000

4000

Speed, rpm

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5000

6000

7000

Power  Power v s. Torque Torque 1.20E+01

1.00E+01

8.00E+00

  s    t    t   a   w  , 6.00E+00   r   e   w   o    P

 P (T )

4.00E+00

=−

2.00E+00

S  NL T  PK 

⋅ T  + S  ⋅ T  2

 NL

0.00E+00 0

0.01

0.02

0.03

Torque, Nm

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0.04

0.05

0.06

Power 

Power vs. Speed 1.20E+01

1.00E+01

8.00E+00

Power vs. Torque

  s    t    t   a   w  , 6.00E+00   r   e   w   o    P

1.20E+01

4.00E+00

1.00E+01

2.00E+00

8.00E+00

0.00E+00

  s    t    t   a   w  , 6.00E+00   r   e 6000   w   o    P

0

1000

2000

3000

4000

Speed, rpm

ω 

1 =

2

5000

7000

4.00E+00

ω max

2.00E+00

0.00E+00 0

0.01

0.02

Peak power is obtained at half of  maximum torque and speed May 21, 2006

0. 03

0.04

0.05

0.06

Torque, Nm

T 

1 =

2

T max

Procedure • Step One: Determine total applied force at worst case • Step Two: Calculate power requirement • Step Three: Calculate torque and speed requirement • Step Four: Find a motor that meets these requirements • Step Five: Plot motor characteristics May 21, 2006

A Few Extra Points

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Simple DC Motor Model V 

=  I  ⋅ R + e

T 

e

= k  ⋅ I  t 

  η max = 1 −    May 21, 2006

 I  NL  I  P 

2

       

= k e ⋅ ω 

V 

=  I  ⋅ R + k  ⋅ ω  e

Motor Inductance • The windings of a DC motor creates an Inductance, L • Change in current through an di V   L inductance creates a voltage dt  • Switching current to a motor causes di/dt to spike (Flyback) =

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Flyback voltages can be very high and damage electronics, that is why a flyback diode in the switching circuit is required.

Winches • Similar to drive motors

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Common Mistakes • Using static or kinetic friction instead of rolling friction  – If a wheel is rolling without slipping, the only energy loss is due to deformations in the wheel/surface (rolling friction)

• Using PWM to control a motor reduces the available torque  – The average power, speed and torque are reduced, however, effective torque is not significantly effected May 21, 2006

Questions?

May 21, 2006