MCE493/593 and EEC492/592

Prosthesis Design and Control

Electromechanical ActuatorsApplications to Prosthetic Devices

Hanz Richter

Department of Mechanical Engineering

2014 1 / 27

Overview

The dynamic equation for a robot was presented as

M(q)q + C(q, q)q + g(q) +

nf∑

i=1

(J i(q))TFi = τ

This assumes direct command of τ , the vector of torques or forces.

An actuator is attached to a subset of the joints to produce τ . The

actuator performs some kind of energy conversion, typically electric to

mechanical, but pneumatic and hydraulic actuators are also common inindustrial robotics. Actuators for precision micromechanical systems

may be piezoelectric.

We will focus on electromechanical (EM) actuators (servo

amplifiers+motors with mechanical transmissions) and their integration

into robotic or prosthetic limb models.

EM actuators are ultimately commanded by power amplifiers, also called

servo amplifiers or servo drives.

2014 2 / 27

Electromechanical actuators

Servo amps may be regenerative (4-quadrant) or not. Commands canbe analog inputs, PWM, serial packets....

Motors may be DC (brushed or brushless, PM or independently-excited),

AC, stepping, linear or rotary. We focus on DC rotary motors.

Many kinds of transmission stages exist: lead/ballscrews, gears (many

kinds), CVTs, rack-pinion...

2014 3 / 27

Electromechanical coupling by magnetic field: Lorentz

force

When a current-carrying conductor is placed in a magnetic field, a forceappears:

F = i(L×B)

i is the current (in A, scalar) L is the length vector (m) and B is the magneticflux density vector (Wb/m2). The force will be in N.

In most electric machines, L ⊥ B, so F = BLi, and in rotating machinesτind = Fr = BLri = αi.

2014 4 / 27

Electromechanical coupling by magnetic field: Induced

voltage

When a conductor moves within a magnetic field, a voltage is induced (the

conductor becomes a small battery):

eind = L.(v ×B)

v is the velocity (m/s). The induced voltage will be in V

In most electric machines, L ⊥ B, so eind = BLv, and in rotating machineseind = BLrw = αw.

2014 5 / 27

Observations

Force (motor effect) and induced voltage (generator effect) coexist in a

machine whenever it is converting power.

The effects constitute a natural negative feedback compensation

satisfying energy conservation.

The distinction between “motor” and “generator” is not fundamental.

Efficiency considerations lead to constructive differences for each.

Constant α is the same for the motor and generator effects. Called

“torque constant”, “motor constant”, “back-emf constant”, “armaturereaction constant”.

2014 6 / 27

Equivalent Circuit and Mechanism

inductor

+

Vin

R

L

i

τL

w

+

eind = αw

τind = αw τfric

DC Machine

To load

With inductance effects, the I/O diff. eq. from Vi to w is second-order:

L

Rw + w +

α2

RJmw =

(

α

RJm

)

Vi −

(

τf + τLJm

)

−L

R

(

τf + τLJm

)

2014 7 / 27

Block diagram representation

Thetawi

T_f+T_L

V_i _

Torque Constant

alpha

Torque Constant

alpha

Integrator

1/s

Integrator

1/s1/Jm

Coil Dynamics

1

L.s+R

The ratio L/R defines an ”electrical time constant”. This will be very small for

motors used in prosthetic applications. Always check!A “mechanical time constant” is defined by considering the no-load velocity

step response, which is determined by friction characteristics. Check

manufacturer’s definition!Example: Maxon EC22 BLDC, 240W: R=3.59 Ω, L=0.63 mH. The time

constant is 175 µs. The mechanical time constant is 2.31 ms (13 timesslower).

2014 8 / 27

Low-order model

We can either model without L from the beginning, make L/R = 0 in the2nd-order ODE or simplify the coil dynamics block to 1/R. We obtain

w +α2

RJmw =

(

α

RJm

)

Vi −

(

τf + τLJm

)

Note that load and friction torque derivatives are not present, which allows us

to use discontinuous friction models.

Sometimes, viscous friction is assumed: τf = bw. Usually, dry (Coulomb)friction is the predominant kind. This can be determined with a simple

power-off, no-load “spin down” test.

2014 9 / 27

Direct parameter measurement procedures

As an alternative to system identification and other parameter estimation

techniques, model knowledge can provide very accurate parameter

measurements.Coulomb friction torque: Apply a small constant current (i) to the motor so

that it moves at a very low constant speed. Under these conditions, theinduced torque is used to balance τf . If α is already known, find τf = αi.Torque constant: Connect motor terminals to an oscilloscope, DAQ or other

high-impedance device so that open-terminal conditions exist for the motor.The motor will be operated as a generator. Use an encoder, tachometer or

other sensor to capture speed. Rotate the motor by hand or with a wrench so

that “good” (high SNR) velocity and voltage curves are captured. Sinceeind = αw, α can be obtained by linear regression.

2014 10 / 27

Direct parameter measurement procedures

Armature resistance and inductance: Use a multimeter or benchtop LCRmeter. Resistance measurement in brushed motors can be difficult from

terminals due to irregular contact between brushes and commutator. Either

remove the brushes and access the commutator or take severalmeasurements. The mode might be more meaningful than the average in this

case.

Rotor inertia and Coulomb friction: Use a power-off spindown test. Readvelocity with a tachometer and oscilloscope or use a DAQ system. Power the

motor to a constant speed and suddenly disconnect power. Motor dynamicsare purely mechanical and become:

w = −τfJm

If τf is a constant (Coulomb), w(t) will be a straight line where the slope is w.

If τf is known, calculate Jm. If τf is unknown, repeat the test with a calibratedinertia ∆J added as a load. Solve a system of 2 equations and 2 unknowns

to find Jm and τf .

2014 11 / 27

Friction models

The study of friction and contact between surfaces is a scientific discipline onits own: tribology. Common models applicable to EM systems:

Viscous friction: τf = bw. Associated with lubricated bearings.Expanded theory: Lubrication and bearing design: MCE567.

Dry or Coulomb friction: τ = fsign(w). Discontinuity may presentdifficulties in some simulations and analyses (can’t differentiate, not an

invertible function). Adequate for w 6= 0.

Stiction (static friction) models: Use deadzones to capture minimum

force required to initiate motion.

LuGre, Dahl and Leuven models: dynamic friction effects.

Comprehensive reference: Armstrong-Hélouvry, B., Control of Machines with

Friction, Springer, 1991.

2014 12 / 27

Real friction, LuGre and other dynamic models

http://people.mech.kuleuven.be/~farid/tribology/friction/macro/properties/transition/transition.html

http://www3.ntu.edu.sg/home/ttegoeh/research.ht

2014 13 / 27

LuGre friction dynamics

z = v − σ0

|v|

g(v)z

Ff = σ0z + σ1z + f(v)

v: relative velocity between surfaces, z: state, Ff : friction force. f(v) is

typically used to capture a viscosity effect: f(v) = σ2v. At steady-state: (show

it!)Ff = g(v)sign(v) + f(v)

g(v) can be chosen to represent Coulomb friction with Stribeck effect:

g(v) = Fc + (Fs − Fc)e−| v

vs|α

2014 14 / 27

Mechanical transmissions

We consider a simple but fairly general 1-dof model for mechanicaltransmissions:

τL

qj

DC Machine

τLTransmission

w

n(qj)qj = w

w

τj

To linkage

Whatever the internal configuration, a transmission will obey a kinematicrelationship

nj(qj)qj = w

where n(qj) is a positive-definite function. Gears, rack-and-pinion, etc. have

n(qj) constant.

The transmission must also obey a force relationship (between τL and τj ).This must be worked out in a case-by-case basis, unless when dealing with

ideal transmissions: τLw = τj qj .Power equality yields the desired force relationship in ideal cases.

2014 15 / 27

Example: Ideal Slider-Crank

Ideal: no mass, no friction.

We show thatFy =

τ

L sin(q/2)

by power equality with τ defined as a loading (opposing q) torque. If the

orientation for τ is chosen as in biomechanics (τ positive when it tries to

produce flexion), a minus sign will be necessary.

2014 16 / 27

Geared transmissions with inertia and friction

τj

qjτL

τf2

To linkage

J2

J1From motor

driving

opposing

w

r2

r1

n =r2

r1

n = DD/CC

Gearing is broadly classified as parallel axes, intersecting axes, crossing

axes.

Kinematics: w = nqjDynamics (j-th actuator)

(

J2 + n2J1)

qj = nτL − τj − τf2

2014 17 / 27

Wormgears

The gear ratio is: number of teeth of gear divided by number of starts(threads) of worm.

Animation:http://en.wikipedia.org/wiki/Worm_drive#mediaviewer/File:Worm_Gear.gif

2014 18 / 27

Rack-and-pinion

τL

w

qj

J1

J2

τf2

τjTo linkager1

Kinematics: w = 1

r1qj

Dynamics (j-th actuator)

(

J2 +J1r21

)

qj = τL/r1 − τj − τf2

2014 19 / 27

Lead/Ballscrews

The lead, l, is the effective travel per unit rotation (m/rad).

2014 20 / 27

Lead/Ballscrews

τf2

qjτLτf1

To linkageJ1

From motor

w

τjJ2

Kinematics: w = 1

lqj

Ideal torque/thrust relationship: T = Fl

Dynamics (j-th actuator)

(

J1 + l2J2)

qj = lτL − lτf1 − l2τj − l2τf2

2014 21 / 27

Lead/Ballscrews: Notes

The ideal torque/thrust relationship is usually modified to include an

efficiency η (ratio of output power to input power).

If the power source is rotary and a linear load is moved, F = ηT/l. If the

power source is linear and a rotary load is moved, T = η′Fl.

For leadscrews, η is near 0.8. For ballscrews a typical value is 0.9.

Actual values depend on load levels, lubrication, contamination, etc.

The efficiency is obtained from manufacturers’ data or from a custom

experiment.

If µ is the friction coefficient between threads and nut, α is the thread

angle and R is the radius, a leadscrew will be self-locking, or

non-backdriveable if

l ≤2πR

cosα

2014 22 / 27

Actuator model integration example

Obtain the kinematic model (overall transmission ratio) and dynamic

relationship between Vi, τ and q:

DC motor:

α,R, Jm, τf

Leadscrew:

η, l, Jl

Gears:

n, J1, J2, τf2 = bw2

w2

M

q

τ

Vi

τf3

2014 23 / 27

Continuously-Variable Transmissions (CVT)

Here n = n(u), where u is an adjustable parameter.

2014 24 / 27

Study: Four-bar linkage with ball/leadscrew and DC

motor

hr

A

B

CD

A

B

CD

d1

d2

H L

a

q3

q3

A

B

T

We develop the kinematic model (L as a function of q3) and relate knee

moment to actuator thrust.

2014 25 / 27

Study: Motor Selection

Assume that the motor will be operating with a specific knee moment

and velocity (from Winter’s data).

Some manufacturers provide speed and thrust specifications: no needto consider motor friction, inertia, etc.

Use moment data to find actuator thrust F . Careful with the signconventions!!

Use knee velocity data to find linear velocity.

Select a motor and test with manufacturer’s force/velocity curves. Check

other limitations: maximum stroke, current, power, etc.

2014 26 / 27

Passive knees

In these devices, F = F (L, L, u). For pure damping action F = b(u)L, whereu is an adjustable parameter. If a spring is included, F = b(u)L+ k(L− L0).We can use the previous study to obtain a model and select how b(u) and kshould be designed, for instance to approximate normal walking.

2014 27 / 27