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2019 Prenscia User Group Meeting | April 30th – May 1st | Novi,MI (USA)

Estimating Rigid Body Motion from Accelerometer Measurements

James Lakowski

Vehicle Dynamics Group

NAVISTAR Inc.

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3Intro to Navistar

Navistar’s share of retail sales in its core markets1 Founded in 1831 as the McCormick Reaper Company

Major manufacturer of commercial trucks, buses and defense vehicles

Headquartered in Lisle, IL

13,200 active employees

Largest dealer network in North America

One of the largest commercial parts distribution networks in North America

1 2015 U.S. and Canada school bus and class 6-8 truck retail sales

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4Product Validation and Analysis @ NAVISTAR

Math Lab Road

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• Accelerometers placed on active and passive sides of engine mounts give localmovement

• How can we get the gross, rigid body motion of the powertrain?

How does the powertrain move when it is mounted in a vehicle?

[video of an engine in a vehicle during a data acquisition goes here]

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𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑐𝑐 + 𝛼𝛼 × 𝑟𝑟𝑐𝑐𝑛𝑛 + 𝜔𝜔 × (𝜔𝜔 × 𝑟𝑟𝑐𝑐𝑛𝑛)

𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑐𝑐 + 𝛼𝛼 × 𝑟𝑟𝑐𝑐𝑛𝑛 + 𝜔𝜔 × (𝜔𝜔 × 𝑟𝑟𝑐𝑐𝑛𝑛)

𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑐𝑐 + 𝛼𝛼 × 𝑟𝑟𝑐𝑐𝑛𝑛

Rigid Body Kinematics

Measured with accelerometers Known from

vehicle geometry

α >> ω (~ 6 orders of magnitude)

Need to estimate (6 unknowns)

Theoretically, point C can be anywhere on the body, but the Center of Mass will be used for this calculation.

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7Rigid Body Kinematics (Contd.)

𝑎𝑎𝑛𝑛𝑥𝑥𝑎𝑎𝑛𝑛𝑛𝑛𝑎𝑎𝑛𝑛𝑛𝑛

=𝑎𝑎𝑐𝑐𝑥𝑥𝑎𝑎𝑐𝑐𝑛𝑛𝑎𝑎𝑐𝑐𝑛𝑛

+𝛼𝛼𝑥𝑥𝛼𝛼𝑛𝑛𝛼𝛼𝑛𝑛

×𝑟𝑟𝑐𝑐𝑛𝑛𝑥𝑥𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛

𝑎𝑎𝑛𝑛𝑥𝑥𝑎𝑎𝑛𝑛𝑛𝑛𝑎𝑎𝑛𝑛𝑛𝑛

=𝑎𝑎𝑐𝑐𝑥𝑥 + 𝛼𝛼𝑛𝑛𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛 − 𝛼𝛼𝑛𝑛𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛𝑎𝑎𝑐𝑐𝑛𝑛 + 𝛼𝛼𝑛𝑛𝑟𝑟𝑐𝑐𝑛𝑛𝑥𝑥 − 𝛼𝛼𝑥𝑥𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛𝑎𝑎𝑐𝑐𝑛𝑛 + 𝛼𝛼𝑥𝑥𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛 − 𝛼𝛼𝑛𝑛𝑟𝑟𝑐𝑐𝑛𝑛𝑥𝑥

𝑎𝑎𝑛𝑛𝑥𝑥𝑎𝑎𝑛𝑛𝑛𝑛𝑎𝑎𝑛𝑛𝑛𝑛

=1 0 0 0 𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛 −𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛0 1 0 −𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛 0 𝑟𝑟𝑐𝑐𝑛𝑛𝑥𝑥0 0 1 𝑟𝑟𝑐𝑐𝑛𝑛𝑛𝑛 −𝑟𝑟𝑐𝑐𝑛𝑛𝑥𝑥 0

𝑎𝑎𝑐𝑐𝑥𝑥𝑎𝑎𝑐𝑐𝑛𝑛𝑎𝑎𝑐𝑐𝑛𝑛𝛼𝛼𝑥𝑥𝑎𝑎𝑛𝑛𝛼𝛼𝑛𝑛

Measured Accelerations (y)Size: N x 1 Geometry (X)

Size: N x 6

Rigid Body Motions (β)Size: 6 x 1

These are the ‘prototype’ rows. Actual order of x,y,z measurements and corresponding rows will depend on transducer setup.

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8Rigid Body Motion (Contd.)

Because N is greater than 6, this system is over constrained. Linear Least Squares Estimation must be used to estimate the beta vector.

Measured Accelerations (known)Size: N x 1

𝑦𝑦 = 𝑋𝑋𝑋𝑋

�̂�𝑋 = 𝑋𝑋𝑇𝑇𝑋𝑋 −1𝑋𝑋𝑇𝑇 𝑦𝑦This estimation will be done every time step.

Component Geometry(known)Size: N x 6 Rigid Body Motions

(unknown)Size: 6 x 1

The Moore-Penrose Pseudo Inverse must be used because X is nonsquare.

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• Each data acquisition can contain up to 120 runs with up to 100 channels per run• Over 20 hours of data sampled at 512 Hz or higher.

• Automating this calculation is necessary

nCode Glyphworks Implementation

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• Step 0: Define the coordinate geometry• Location of the transducers used for measuring acceleration on the rigid body• Defined from physical measurements or CAD models

• An Excel macro generates a .xml metadata file containing the geometry data for each acceleration measurement being used for the rigid body motion calculation

nCode Glyphworks Implementation

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11nCode Glyphworks Implementation

Raw time histories

Channels to be used for rigid body calculation

Metadata containing x,y,zcoordinates of transducers.

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12nCode Glyphworks Implementation

Associate the component geometry with the raw data

Calculate Rigid Body Motion!

Bandpass filter to remove high frequency content and DC offset

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• Written with the Python Scripting Glyph

• Originally developed in MATLAB• LAPack is LAPack

• Python does not require a license

• Buffered processing for speed• np.pinv() for numerical stability

• Can accept an arbitrary number of channels and accelerometer orientations

nCode Glyphworks Implementation

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• If we are able to accurately estimate the rigid body accelerations, then acceleration at any point on the body can be calculated too.

𝑎𝑎𝑣𝑣 = 𝑎𝑎𝑐𝑐 + 𝛼𝛼 × 𝑟𝑟𝑐𝑐𝑛𝑛

• DANGER! This method will not include motion due to local deflection or excitation!• Local bending mode• Powertrain excitation / forced response

Use Cases - Virtual Accelerometers

EstimatedKnown from component geometry

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• Measurements from 4 triaxial accelerometers at the engine mount locations were used for this calculation

• A 5th triax was installed on the top of the flywheel housing in between the engine and the transmission.

• This triax was not used for the rigid body motion calculation. • A virtual acceleration calculation was performed for the flywheel housing

accelerometer using the rigid body motion estimates from the engine mount measurements.

• The results are plotted in the following slides.

6DoF Peformance

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166DoF Performance

Calculated using 6DoF and Virtual Acceleration

Measured

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176DoF Performance

Calculated using 6DoF and Virtual Acceleration

Measured

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186DoF Performance

Pretty good

Calculated using 6DoF and Virtual Acceleration

Measured

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196DoF Performance

Correct frequency content, but amplitudes are a bit low…

Calculated using 6DoF and Virtual Acceleration

Measured

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206DoF Performance

ChanTitle Max Min Range % difference RMS % differenceT_512@TCAX1_C74.RN_1 0.72 -0.71 1.44 -40.73 0.08 -44.63Trans_Housing X 0.45 -0.41 0.85 0.05

T_512@TCAY1_C75.RN_1 0.86 -0.87 1.73 -55.30 0.08 -41.87Trans_Housing Y 0.37 -0.40 0.77 0.04

T_512@TCAZ1_C76.RN_1 0.96 -0.83 1.78 -30.87 0.11 -17.99Trans_Housing Z 0.60 -0.63 1.23 0.09

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21Use Cases – Powertrain Motion Design Iterations

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22Use Cases – Cab Motion

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• Estimated Powertrain 6DoF from accelerometer measurements to identify coupling between axle motion and powertrain rigid body modes.

Use Cases – Wobble

Powertrain Roll/Yaw response lags the axle input by about 90 Degrees which is indicative of a driven resonance.

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• Low amplitude / directional amplitude contrast

Limitations

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Challenge Solution

Summary

Results

Determine the rigid body motion of vehicle components such as powertrain, cab, cooling module, etc.

Using some basic dynamics, linear algebra and python programming, this can be added to our standard data reduction processes in nCode.

More fundamental information regarding the performance of suspension systems can be determined in roughly the same amount of time

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