Contact Dynamics ExperimentModelling

A Report Submitted in Partial Fulfillment

of the Requirements for SYDE 652

Mike Boos

Faculty of Engineering

Department of Systems Design Engineering

April 16, 2009

Course Instructor: Professor John McPhee

Abstract

A contact dynamics model has been proposed that uses the properties of the volume

of interference between to solid geometries to determine contact forces and torque. A

series of potential experiments are described for measuring the contact normal and

friction model parameters and for model validation. The experimental apparatus are

simulated using the volumetric model and results are compared with simulations in

ADAMS, where possible.

i

Contents

Abstract i

Contents iii

List of Tables iv

List of Figures v

1 Introduction 1

1.1 Contact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Experiments 3

2.1 Contact normal experiments . . . . . . . . . . . . . . . . . . . . . . . 4

Volumetric stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Friction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Translation and rotation . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Modelling 8

3.1 Payload geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Spherical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Cylindrical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Contact normal experiments . . . . . . . . . . . . . . . . . . . . . . . 9

ii

3.3 Friction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Translation and rotation . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 ADAMS models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Simulation results 12

4.1 Contact normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Quasi-static loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Translation and rotation . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Conclusions 15

Appendix A - Friction model 17

Appendix B - Model parameters 19

iii

List of Tables

1.1 Model parameters requiring experimental identification. . . . . . . . . 2

iv

List of Figures

2.1 Experimental apparatus for measuring displacement and contact force. 4

2.2 Apparatus for friction experiments, viewed in plane of motion. . . . . 5

2.3 Apparatus for friction experiments, viewed off-angle. . . . . . . . . . 5

4.1 Displacements for a constant loading rate of 1 N/s. . . . . . . . . . . 12

4.2 Results of damping experiments for x0 = 0 and Factuator = 20. . . . . 13

4.3 Results of damping experiments for x0 = 0.02 and Factuator = 20. . . . 13

4.4 Results of friction in pure translation experiments. . . . . . . . . . . . 13

4.5 Results of friction in pure rotation experiments. . . . . . . . . . . . . 14

4.6 Results of combined translation and rotation experiments. . . . . . . 14

v

1

Introduction

A flexible volumetric contact dynamics model has been proposed for the purpose of

generating reliable simulations of space-based manipulator contact dynamics rapidly

[1]. In this model, forces and moments between two bodies in contact can be expressed

in terms of the volume of interference between the undeformed shapes of the bodies.

This model is to be validated experimentally for hard-on-hard (i.e.: metal on

metal) contact. Additionally, in order to use the model to simulate actual contact

scenarios, model parameters must be determined experimentally.

A series of contact experiments have been proposed. These experiments include

contact normal measurements to estimate and validate stiffness and damping parame-

ters, as well as experiments involving tangential and rotational motion to find friction

forces and spinning resistance torque.

It is the purpose of this work to present dynamic models of these experiments in

order to later validate the volumetric contact model against experimental results.

The experimental apparatus for these experiments are pending construction, so

the models have yet to be validated empirically. However, the experiments have been

modelled in ADAMS, where possible, and compared with the simulation results from

the mathematical models.

1

1.1 Contact model

The volumetric model presented by Gonthier [1] is briefly described below.

In this model, the contact normal force is related directly to the size of the volume

of interference, V , through a volumetric stiffness kV , given in units of force per unit

volume. The magnitude of the normal force is then given by

FN = kV V (1 + avcn) (1)

where vcn is the speed of the centroid of the volume in the normal direction and a is

damping parameter determined by the coefficient of restitution and the initial impact

velocity.

Friction is modelled through a bristle friction force model. In addition, friction

torque that resists spinning in the normal direction is also modelled. The model also

attempts to account for the Contensou effect, where tangential friction effects can be

diminished when there is spinning motion in the normal direction. Details of this

friction model are included in Appendix A.

Parameters for the model requiring experimental identification are listed in Ta-

ble 1.1.

Table 1.1: Model parameters requiring experimental identification.

Parameter Description

kV Volumetric stiffnesseeff Coefficient of restitutionvsmall Minimum damping velocity thresholdµS Stiction friction coefcient (static friction)µC Coulomb friction coefcient (kinetic friction)σ0 Load-dependent bristle stiffnessσ1 Load-dependent bristle damping coefcientσ2 Load dependent viscous damping coefcientvS Stribeck velocityτdw Dwell-time dynamics time constant

2

2

Experiments

The purpose of the proposed experiments is to determine parameters for the volu-

metric contact model and validate that model. Parameters to be determined include

a volumetric stiffness constant, damping parameters, and friction parameters. Of in-

terest for validation are the volume-normal force relationship, bristle-friction model,

spinning friction torque model, and the Contensou effect. This chapter describes the

proposed physical experimental apparatus for parameter identification and validation,

as well as the procedure for driving the bodies in the experiments.

For the contact normal force model, measurements of the displacement and forces

in the normal direction are desired. For tangential friction, the contact translational

displacement and friction forces must be measured, and for spinning friction, the

normal angular velocity and spinning friction torque must be measured.

Ideally, one should control and measure all these displacements and forces directly

and simultaneously. To accomplish this, three actuators would be required. However,

the experiments can be divided into two types, contact normal and friction. This re-

duces the number of actuators required at any given time to one and two respectively.

Additionally, identifying model parameters is easier if each set of parameters can be

measured independently.

Two different contact payloads are proposed. The first is spherical, and the results

from volumetric contact simulation can be compared directly with those from Hertz

theory. The second is cylindrical, with a flat end forming one of the contact surfaces

3

with the opposing plane. This payload does not satisfy the assumptions in Hertz

theory of a small contact patch and non-conforming geometries, which allows testing

of the volumetric model in cases where Hertz theory does not apply.

2.1 Contact normal experiments

The contact normal experiments will be controlled with a single ball screw-type linear

actuator (which allows for some back-driving). The contact payload will be mounted

rigidly onto the actuator carriage and driven into a metal plate, as shown in Figure 2.1.

One or more normal force sensors mounted behind the plate will measure the reaction

force, while the encoders of the motor driving the linear actuator will determine the

displacement of the payload.

Figure 2.1: Experimental apparatus for measuring displacement and contact force.

Volumetric stiffness

Starting from rest, with the payload touching the contact surface with no forces

between them, the force driving the payload will gradually be increased from zero so

that the force sensors are loaded quasi-statically. The increase in displacement should

be gradual such that the effect of damping is negligible. The measured displacement

can be used to find the volume of interference, so that a volumetric stiffness constant,

kV , can be estimated through a linear fit of force to volume measurements.

Damping

Damping depends on the initial normal velocity at impact, vin, as well as a coefficient

of restitution, eeff . With a known volumetric stiffness and measured displacements,

velocities, and forces, the coefficient of restitution can be estimated.

The coefficient should be estimated using a variety of higher initial impact veloc-

ities and driving forces. In addition, the payload should be driven at lower impact

4

velocities in order to estimate vsmall, the minimal velocity threshold. The system

could be driven with a near-zero initial impact velocity to estimate vsmall or a se-

ries of experiments could be conducted starting from a near-zero initial velocity and

increasing the initial velocity until the measured damping parameter a changes.

If back-driving of the ball screw is possible, the initial and final impact velocities

should be compared with the coefficient of restitution.

2.2 Friction experiments

The friction experiments require an apparatus that controls both tangential motion

and normal rotation, in addition to applying a contact normal force. The apparatus

depicted in Figures 2.2 and 2.3 has been designed to accommodate this. A linear

actuator (consisting of a DC brushless motor connected to a ball screw) drives the

translational motion, while a small brushless DC motor drives rotation. The payload

is mounted to the shaft of the small motor, the frame of which is mounted through

a vertical linear guide to the carriage of the linear actuator to permit free motion in

the normal direction.

As the system is under gravity, the normal force on the payload is defined by

the masses of the motor and payload. Since the stiffness constant is estimated from

the contact normal experiments, the properties of the volume of interference can be

determined.

Two 3-DOF (x,y,z) force sensors beneath the contact plate connect it to the

ground. These are aligned so that the sensors are centred in the plane of motion

of the payload. The normal force can be measured through the sum of the y-forces,

the tangential friction force through the sum of the x-forces, and the spinning friction

torque through the difference of the z-forces multiplied by the distance between them.

Figure 2.2: Apparatus for friction experiments, viewed in plane of motion.

Figure 2.3: Apparatus for friction experiments, viewed off-angle.

5

Alternative designs were considered. A four-axis CNC would enable the desired

motion, however, concerns over control and real-time position measurement made

such a choice unfeasible. In addition, many CNC designs would require the contact

plate to be moved to enable relative motion with respect to the payload, which would

register on the force sensors. Parallel manipulators were also considered, but were

considered to be limited in their range of motion.

Translational motion

The purpose of using pure translational motion is to determine the seven bristle

friction model parameters and to validate that model for the surfaces in contact.

To find the coefficient of static friction, µs, the applied force can be increased

until the payload begins to move. The coefficient of static friction is the peak friction

force measured at the instant before movement divided by the contact normal force.

This should be performed at several different applied normal loads for a more reliable

estimate [1].

The Coulomb friction coefficient, µc and viscous damping coefficient, σ2 can be

determined through experiments where the payload is driven at various different

constant velocities [1].

If dwell-time dependency is of interest, the payload must be allowed to come to

a stop and allowed to dwell for a period of time and then be forced to move again.

By varying the time that the payload dwells, the dwell-time dynamics time constant,

τdw, can be estimated.

Gonthier [1] provides suggestions as to how to find the bristle stiffness and damping

parameters, which may be difficult. These values, along with the Stribeck velocity

will likely need to be determined through parameter tuning from experiments where

the payload is forced to enter into slipping from rest.

6

Rotational motion

The spinning friction torque model uses the same parameters as those determined

by the translational experiments. The main purpose of the rotational experiments is

to validate this torque model. Thus, similar experiments can be applied where the

payload is rotated instead of translated to determine if the model fits the data.

Translation and rotation

The purpose of these experiments are to validate the model’s description of the Con-

tensou effect.

One type of experiment would be to vary rotational speed while holding the tan-

gential speed constant and measuring the resulting friction forces and torques. Con-

versely, rotational speed could be held constant with varying tangential speeds. The

model Contensou factors could then be compared with the measured ones.

7

3

Modelling

3.1 Payload geometries

Spherical

Given a sphere-plane contact with a radius r, and a depth of penetration δ [2], the

volume of interference is:

V =1

3πδ2(3r − δ) (1)

and the volume moment of inertia about the normal axis [3] is

JV,nn =2

5V r2 +

2

5V

(r − δ)(r − δ2)2

r − δ2

(2)

The radius of the spherical payload was defined as 5 mm.

Cylindrical

Given a cylinder-plane contact with a radius r, and a depth of penetration δ, the

volume of interference is:

V = πδr2 (3)

8

and the volume moment of inertia about the normal axis is

JV,nn =1

2r2V (4)

The radius of the cylindrical payload was defined as 5 mm.

3.2 Contact normal experiments

The contact normal apparatus has one single degree of freedom, in x. The equation

of motion is given by

mx = Factuator − FN (5a)

= Factuator − kV V (1 + ax) (5b)

The mass of the payload and carriage is 4.65 kg.

For quasi-static stiffness experiment, the payload begins at rest just touching the

plate, and the applied force is increased at a constant rate from 0 to 30 N over 30 s.

For the damping parameter estimation experiments, the payload begins just touch-

ing the plate with a given initial velocity and a constant applied force. Back-driving

of the ball screw is permitted. Modelled are two examples having initial velocities of

0 and 20 mm/s, both with an applied force of 20 N. The system was simulated for

0.1 s.

Simulation of equation (5b) was performed using the ode45 solver in MATLAB.

3.3 Friction experiments

The friction experiments have three degrees of freedom: the translation of the linear

actuator in x, the vertical displacement of the payload in y, and the rotation θ of

the payload about the y-axis. The equations of motion have been reduced to the

9

following three differential equations:

(mpayload +mmotor +mcarriage)x = Ff + Factuator (6)

(mpayload +mmotor)y = FN − (mpayload +mmotor)g (7)

Ipayloadθ = Ts + Tmotor (8)

Throughout the experiments, y is kept at equilibrium, so (7) becomes

FN = (mpayload +mmotor)g (9)

Also note that the friction model introduces three new state variables for trans-

lational and rotational bristle deformation and dwell time. Differential equations for

these variables and formulas for tangential friction and spinning resistance torque are

given in Appendix A.

To solve the six differential equations, the ode15s solver in MATLAB was found to

be necessary to simulate some of the experiments. Parameters used in the experiment

models are listed in Appendix B.

Translation

The implemented simulations involve a constant tangential load being applied. Two

different loads have been simulated, 10 N and 20 N, which lead to sticking and sliding

respectively.

Rotation

The implemented simulations involve a constant rotational torque being applied. Two

different loads have been simulated, 0.025 Nm and 0.075 Nm which lead to sticking

and sliding respectively.

10

Translation and rotation

The proposed experiments involve prescribed motions, so only the friction state dy-

namic equations need to be solved in order to determine the friction forces and torque.

The first simulation entails a constant tangential speed of 0.1 m/s and a rotational

speed increasing at a constant rate from 0 to 200 rad/s over 5 s. The second has a

constant rotational speed of 100 rad/s and a tangential speed increasing at a constant

rate from 0 to 0.2 m/s over 5 s.

3.4 ADAMS models

The two experimental apparatus were constructed in ADAMS for comparison with

the mathematical model. ADAMS was configured to use the IMPACT model, which

treats contact like a non-linear spring-damper, based on depth and speed of pene-

tration, and includes a Coulomb friction model. Contact parameters are listed in

Appendix B. The GSTIFF solver was used for dynamic simulation.

The contact stiffness model required a stiffness parameter and a force exponent

to apply to the penetration. For the spherical payload, the contact stiffness could be

modelled after Hertz theory. For the cylindrical payload, a linear relationship was

given for contact stiffness, in keeping with the apparent linear relationship between

normal force and penetration in the volumetric model.

Contact damping parameters were more difficult to correlate with other theoretical

models. A maximum damping value is required, as well as some penetration depth

to apply it at. This is implemented using the cubic ‘STEP’ function. An attempt

was made have the damping/penetration profile be similar to that for the spherical

payload at low impact velocities, however the shape of the STEP function makes for

a poor fit. This was found to lead to poor correlation in the results.

IMPACT provided a simple Coulomb friction model for tangential friction only,

where the coefficient of friction is a function of slip velocity. The model does not

account for spinning friction torque or the Contensou effect. Thus, for the friction

models, only the translational friction experiments could be compared.

11

4

Simulation results

4.1 Contact normal

Quasi-static loading

Figure 4.1 shows displacements for the spherical and cylindrical payloads as the ap-

plied force is increased from 0 to 30 N. For the spherical payload, the volumetric

model’s displacement profile differs from the Hertz theory model in ADAMS, as they

have different ordered relationships between force and displacement. For the cylindri-

cal payload, the volumetric and ADAMS models share the same force-displacement

relationship.

Figure 4.1: Displacements for a constant loading rate of 1 N/s.

Damping

Comparisons between the volumetric and ADAMS models for the damping parameter

experiments are given in Figures 4.2 and 4.3. Note that back-driving of the linear ac-

tuator is allowed, permitting the oscillations shown. It is apparent that the difference

between the damping models in the volumetric models and ADAMS lead to signifi-

cant differences in performance, especially for the cylindrical payload, which has an

12

initially steeper rate of loading. For higher velocity impacts shown in Figure 4.3, the

damping has little effect on the ADAMS model.

Figure 4.2: Results of damping experiments for x0 = 0 and Factuator = 20.

Figure 4.3: Results of damping experiments for x0 = 0.02 and Factuator = 20.

Convergence of simulation results in the volumetric models were found using a

relative solver tolerance of 1 × 10−6, while convergence for the ADAMS model was

found using tolerances around 1×10−4 and 1×10−11 for the spherical and cylindrical

payloads, respectively. The tolerance required for the cylindrical payload suggests

that the equations of motion are very stiff in ADAMS and that model could use

additional damping to stabilize after impact.

4.2 Friction

Translation

For friction in pure translation, the model is identical for both payload geometries.

Results from the two modelled experiments are given in Figure 4.4. The bristle friction

model allows for greater displacement in static friction, through the bristle stiffness,

while the ADAMS model has some gradual creep since friction is a function of slip

velocity, as shown by the 10 N case. From the 20 N case, the forces are great enough

to overcome static friction and payload is in motion, so the friction force reaches

a steady state. Convergence of simulation results for the volumetric model occurs

Figure 4.4: Results of friction in pure translation experiments.

for relative tolerances below 1 × 10−6, while the ADAMS model converges around

1× 10−4. Note that very low stiction and friction transition velocities are required

13

Rotation

With rotation, the friction models for the payloads differ, as they depend on the

properties of the volumes of interference. Results for applied torque of 0.025 N and

0.075 N are shown in Figure 4.5. For the sticking case, both payloads experience a

slight initial impulse as the rotational bristles deform. For the sliding case, the spher-

ical payload experiences greater friction torque as the normal pressure distribution

is focused intensely around the point of contact, where the surface velocity is small.

The cylindrical payload has a normal pressure distribution that is evenly distributed

over a larger surface area, so there is less friction at the slow moving points toward

the centre.

Figure 4.5: Results of friction in pure rotation experiments.

Translation and rotation

Results from the combined translation and rotation experiments are shown in Fig-

ure 4.6. The left column has angular velocity increasing at a constant rate while the

right column has tangential velocity increasing at a constant rate. Notice that the

magnitude of the friction force decreases as the angular velocity increases, and the

magnitude of the spinning friction torque decreases as tangential velocity is increased.

Figure 4.6: Results of combined translation and rotation experiments.

14

5

Conclusions

A series of experiments have been proposed for parameter identification and validation

of a volumetric contact dynamics model. Two experimental apparatus are presented

for measuring the contact normal parameters of volumetric stiffness and damping,

and friction model parameters. Friction experiments are divided into pure tangential

translation, pure normal rotation, and combined translation and rotation.

The experimental apparatus were modelled through dynamic equations. Simula-

tions of these equations were compared with results from models of the experimental

apparatus in ADAMS, using the built-in contact model. The ADAMS friction model

was limited to translational friction, so simulations of experiments with rotational

motion could not be compared with the ADAMS models. In addition, the damping

model in ADAMS was radically different from that of the volumetric model, so it was

difficult to compare the damping experiment simulations.

The dynamic equations used to model the experimental apparatus will be useful

in simulating the experiments for the identification of parameters and validation of

the volumetric contact dynamics model.

15

References

[1] Y. Gonthier, Contact Dynamics Modelling for Robotic Task Simulation. PhD

thesis, University of Waterloo, 2007.

[2] E. W. Weisstein, “Spherical cap.” From MathWorld–A Wolfram Web Resource.

http://mathworld.wolfram.com/SphericalCap.html.

[3] D. Gauchez and J. Souchay, “Simulation of post-impact rotational changes

through multi-dimensional parametrization,” Icarus, vol. 185, no. 1, pp. 83 –

96, 2006.

16

Appendix A - Friction model

A brief description of the equations for the friction model are described. For deriva-

tions, see [1].

For the bristle friction force model, a new state vector z is defined to represent

bristle deformation. The bristle deformation rate is given by

z = svsct + (1− s)( 1

σ1µcdirε(vsct, vε)Cv,s −

σ0σ1

z) (1)

where vsct is the tangential velocity at the centroid of the contact surface. The

deformation rate is saturated, such that

z =1

σ1sat(σ0z + σ1z, µmaxCv,s)−

σ0σ1

z (2)

The s value determines whether sticking is occurring and is defined by

s = e−

v2avg

v2s (3)

where

v2avg = vsct · vsct + ωnJV,nnV

ωn (4)

ωn is the angular velocity in the normal direction and JV,nn is the volume moment of

inertia about the normal.

Similarly, a bristle state θn can be defined for rotation. The deformation rate is

θn = sωn + (1− s)(µcCω,sσ1rgyr

sgn(ωn)− σ0σ1θn (5)

17

The spinning deformation rate is also saturated:

θn =1

σ1sat(σ0θn + σ1θn,

µmaxCω,srgyr

)− σ0σ1θn (6)

The radius of gyration, rgyr is defined by

r2gyr = JV,nn/V (7)

The Contensou dimensionless factors Cv,s and Cω,s are given by

Cv,s = s+ (1− s) |vsct|vavg

(8)

Cω,s = s+ (1− s)rgyr|ωn|vavg

(9)

A final state variable, sdw is introduced to describe the dwell time of the system

for the maximum stiction force

sdw =

1τdw

(s− sdw) s− sdw ≥ 0,

σ0σ1

s− sdw < 0(10)

From this dwell time, the maximum coefficient of friction can be determined:

µmax = µc + (µs − µc)sdw (11)

Finally, the tangential friction force and the spinning friction torque are given by

Ff = −FN(sat(σ0z + σ1z, µmaxCv,s) + σ2vsct) (12)

τs = −r2gyrFN(sat(σ0θn + σ1θn,µmaxCω,srgyr

) + σ2ωn)n (13)

18

Appendix B - Model parameters

Table B.1 Mathematical model parameters.

Parameter Value Units

eeff 0.85 -

vs 1.00 mm/s

vε 0.10 mm/s

vsmall 1.00 mm/s

µS 0.8 -

µC 0.4 -

σ0 105 1/m

σ1√

105 s/m

σ2 0 s/m

τdw 0.75 s

g 9.81 m/s2

mpayload 0.083 kg

mmotor 2.88 kg

mcarriage 6.62 kg

Ipayload 6.45 kg ·mm2

19

Table B.2 ADAMS model parameters.

Parameter Value Units

Spherical

k 3.77× 103 N/mm3/2

e 1.5 -

d 4.6 N/(mm/s)

c 4.0× 10−2 mm

Cylindrical

k 9.82× 104 N/mm

e 1 -

d 4.6 N/(mm/s)

c 4.0× 10−2 mm

20