Online Calibration of Vehicle Powertrain and Pose EstimationParameters using Integrated Dynamics

Neal Seegmiller, Forrest Rogers-Marcovitz, and Alonzo Kelly

Abstract— This paper presents an online approach to cal-ibrating vehicle model parameters that uses the integrateddynamics of the system. Specifically, we describe the identifi-cation of the time constant and delay in a first-order modelof the vehicle powertrain, as well as parameters requiredfor pose estimation (including position offsets for the inertialmeasurement unit, steer angle sensor parameters, and wheelradius). Our approach does not require differentiation of statemeasurements like classical techniques; making it ideal whenonly low-frequency measurements are available. Experimentalresults on the LandTamer and Zoe rover platforms show onlinecalibration using integrated dynamics to be fast and moreaccurate than both manual and classical calibration methods.

I. INTRODUCTION

Accurate calibration is essential for high-performance con-trol and position estimation on autonomous vehicles [1].

The required calibration includes a model of the power-train dynamics. For a wheeled mobile robot (WMR), thepowertrain is the group of components that generate powerand transmit it to wheels. No powertrain is capable ofinstantaneously changing wheel speeds; often delays andtransients in the powertrain response are significant and mustbe accounted for using model predictive control.

The calibration also includes intrinsic parameters requiredfor position estimation. If the position estimation systemincludes an inertial measurement unit (IMU), the pose ofthe IMU relative to the vehicle body frame must be known.Wheel odometry requires accurate dimensional values suchas wheelbase and wheel radius, as well as calibrated steerangle sensors for articulated wheels.

Unfortunately, manual calibration of all these parameterscan be time-consuming and inaccurate. For example, theIMU is often mounted deep in the robot chassis, makingposition offsets difficult to measure. Nominal CAD dimen-sions often can not be trusted. Furthermore, calibration mustbe repeated as hardware degrades or shifts with use in thefield. Ideally, vehicle model calibration should be automated,online, and accurate.

A. Related Work

Many WMR controllers ignore powertrain dynamics andrely on feedback to correct errors that could be preventedwith better predictive models. Others operate at low orconstant speeds where the effects of powertrain transients

Manuscript received September 16, 2011.N. Seegmiller is a Ph.D. candidate at the Carnegie Mellon University

Robotics Institute, F. Rogers-Marcovitz is a robotics engineer at the NationalRobotics Engineering Center, and A. Kelly is a professor at the RoboticsInstitute {nseegmiller,forrest,alonzo}@cmu.edu

are negligible. Yu et al. developed a dynamic skid-steeredvehicle model that accounts for motor saturation and powerlimitations, but do not attempt automated calibration [2].Others have devised methods of identifying time delays inthe context of sensor calibration, e.g. for LIDAR [3] and forIMU/camera systems via a registration technique [4].

Outside of robotics literature, there are methods of cali-brating induction motor parameters [5] and engine modelstoo complex for our needs. However, we are aware of nounified method of simultaneous online calibration of a timeconstant and delay for a WMR powertrain, much less withthe robustness to noise of our presented approach.

Recent work on calibrating IMU position has focused onthe relative pose between the IMU and other sensors [6][7], not between the IMU and vehicle body frame. Somehigh-end GPS/INS systems such as the Novatel SPAN haveproprietary calibration routines for the IMU to GPS antennaoffset and angular offsets with respect to the vehicle, butthese require high-frequency differential GPS updates1.

An early example of automated odometry calibration isthe “UMBmark” test in which wheelbase and wheel diam-eter are identified as the WMR traverses a preprogrammedsquare trajectory [8]. Roy and Thrun presented an online,probabilistic method of calibrating odometry, but it requiresfrequent sensor scans and learns simplified translational androtational error parameters [9]. Antonelli et al. present a cal-ibration process similar to ours in that predicted trajectoriesare integrated, but theirs is an offline process that learnsa redundant set of parameters [10]. We are aware of nomethod that simultaneously calibrates both IMU offsets andodometry parameters online using temporally sparse posedata.

B. Integrated (Perturbative) Dynamics

This paper presents an integrated equation error approachto calibrating intrinsic vehicle model parameters that usesthe integrated dynamics (ID) of the system. System modelsare commonly known in the form of a differential equation:

x = f(x, u, p) (1)

where x is the state vector, u is the input vector, and pis the vector of parameters to be identified. The classicalmodel identification approach uses the differential equationdirectly, which requires observations of x and x. Often xcan not be measured directly, so measurements of x are

1SPAN-SE User Manual, http://www.novatel.com/assets/Documents/Manuals/om-20000124.pdf

numerically differentiated with respect to time. For example,the Springer Handbook of Robotics teaches the estimation ofmanipulator inertial parameters using this approach, whichrequires double differentiation of joint angle data [11].

In contrast, we identify p using the integrated dynamics:

x(t) = x(t0) +

∫ t

t0

f(x(τ), u(τ), p)dτ (2)

In effect, we integrate the prediction rather than differentiatethe measurement, which has several advantages. This ap-proach avoids numerical derivatives which can be noisy andrequire accurate timestamps and high-frequency sensors. Inaddition, integrated predictions account for delayed effectsof the parameters. For example, angular velocity error hasno instantaneous effect on translational velocity, but doeshave an increasing effect on translational position error withdistance traveled.

The cost of using integrated dynamics is added complexityin deriving the Jacobian (or gradient) required for parameterestimation. ∂x(t)

∂p is harder to compute than ∂x∂p because it

requires at least a numerical solution to the (often non-linear)differential equation. To facilitate efficient computation of theJacobian in our nonlinear WMR application we derive theintegrated perturbative dynamics (IPD) of the system. Theseare obtained by solving the linearized differential equation:

δx = Fδx+Gδu, F =∂f

∂x, G =

∂f

∂u(3)

δx(t) = Φ(t, t0)δx(t0)

∫ t

t0

Φ(t, τ)G(τ)δu(τ,∆p)dτ (4)

(3) and (4) are valid linear approximations of the systemfor small perturbations δx and δu. Here, we chose to limitthe parameterization to the input vector u; perturbations δuare attributed to errors in the parameter estimates ∆p. Thetransition matrix Φ is explained in Section III-C. Previously,the authors demonstrated the use of IPD for the identificationof slip parameters and stochastic dynamics [12]; here IPD isused to identify intrinsic odometry parameters.

For brevity, we will refer to our integrated equationerror approach as the “ID/IPD approach” and the classicalapproach (which uses the differential equation) as the “DEapproach.” In the following sections we explain how ID/IPDcan be used with an extended Kalman filter (EKF) toestimate vehicle parameters online. Section II explains theidentification of a time constant and delay in a powertrainmodel, and Section III explains the calibration of IMUposition offsets, steer angle sensors, and wheel radius. Ex-perimental results show improved convenience and accuracyover manual calibration. Note that no special trajectoriesare required; calibration can be performed during normaloperation. Because identification is performed online, thecalibration should automatically adjust to changing hardwareconditions (e.g. a low battery or a flat tire).

II. CALIBRATION OF THE VEHICLE POWERTRAIN

WMR powertrains can not change wheel speeds instanta-neously; doing so would require infinite torque. The dynam-

ics of many vehicle powertrains can be modeled by a timedelay and a first-order transient response:

ω =1

τc(ω(t− τd) − ω(t)) (5)

where ω denotes the angular velocity of the wheel. Wepresent a model for an individual wheel, but models mayalso be learned for sets of wheels. ω denotes the commandedvelocity, τc the time constant, and τd the time delay. Giventhe angular velocity at some initial time t0 the velocity atthe future time t is given by the integral:

ω(t) = ω(t0) +

∫ t

t0

1

τc(ω(τ − τd) − ω(τ)) dτ (6)

In practice this integral is computed numerically using therecursive discrete-time relation:

ω[i+ 1] = ω[i] +1

τc

(ω[i− τd

∆t] − ω[i]

)∆t (7)

where the integer in brackets [ ] denotes the time index and∆t denotes the time step size.

A. Online Identification of Powertrain Parameters

The time constant τc and time delay τd can be estimatedonline using an extended Kalman filter. Here we identifyall the necessary variables to perform an EKF measurementupdate: the state x, measurement z, predicted measurementh(x), measurement uncertainty R, and the measurementJacobian H . In short, at time t0 we predict the wheel velocityat some future time t; at time t we compare the measuredwheel velocity to the prediction, and update our estimates ofτc and τd accordingly.

The EKF state contains the parameters we are estimating:x =

[τc τd

]T. The measurement z is the wheel velocity

at time t, as measured by the wheel encoder. The predictedmeasurement h(x) is the predicted wheel velocity at time tand is calculated using (6) and our current estimates of τcand τd. The measurement uncertainty R is sensor-dependent.

The measurement Jacobian is: H =[∂ω(t)∂τc

∂ω(t)∂τd

]. The

partial derivative with respect to the time constant τc is:

∂ω(t)

∂τc=

∫ t

t0

−1

τ2c(ω(τ − τd) − ω(τ)) − 1

τc

∂ω(τ)

∂τcdτ (8)

This partial derivative is also calculated numerically using arecursive, discrete-time relation analogous to (7). No straight-forward analytical solution for the derivative with respect tothe time delay is known, so ∂ω(t)

∂τdis calculated numerically

(using the central-difference for example).

B. Experimental Results

Experimental results were obtained on a LandTamer, ahydraulic-drive skid-steered autonomous vehicle (see Fig. 1).As the left and right wheels exhibited similar dynamics, asingle model was learned for both. A time interval (t − t0)of 0.5 seconds was chosen; the time interval should be onthe order of the time constant τc.

In the first test, the LandTamer autonomously drove incircular trajectories of varying curvature and speed. In Fig.

Fig. 1. The LandTamer, a hydraulic-drive skid-steered autonomous vehicle.

TABLE IPOWERTRAIN MODEL MEAN SQUARED ERROR (DEG/S)2

Calibration methodTest Command DE ID

Circles 70.97 11.78 9.92Teleop 501.37 105.20 94.53

2 the vehicle is commanded to instantly decelerate whiletraversing an arc. The powertrain model, using the onlineintegrated dynamics estimates of the time delays and timeconstants, accurately captures the transient response of theactual wheel speeds (as measured by encoders).

In the second test, the LandTamer is teleoperated andcommands alternate between ∼ 0.5 and 4 rad/s (see Fig.3). The commands change so quickly that the wheels rarelyreach their target speed, which is accurately captured by thecalibrated model. Note also that the parameters learned inthe second test differ from those in the first; this suggeststhe powertrain dynamics depend on the operating mode,which justifies the need for online calibration. The rate ofconvergence depends on the variation in velocity commands.In this test, Fig. 4 shows convergence in one minute startingfrom poor initial estimates.

We also implemented an online identification filter thatuses the differential equation directly. Every effort was madeto improve performance, including careful tuning of a bandpass filter for the noisy double derivatives of encoder ticks(i.e. the measurement ω), but the classical DE approach stillunderperformed the ID approach, especially at identifyingthe time delay. This is because, using the DE approach, ∂ω

∂τdis

observable (or nonzero) only at the instant when commandschange but using the ID approach, ∂ω(t)

∂τdis observable

whenever a command change occurs in the interval t − t0.Intentionally poor initial parameter estimates were chosen(τc = 0.2 s, τd = 0.25 s), but with sufficiently large initialuncertainty, to highlight the difference in performance.

Table I presents the mean squared error (MSE) of wheelvelocity predictions for powertrain models calibrated usingboth the DE and ID approaches, and one that naıvely predictsthe nominal commands. MSE is larger in the “Teleop” testbecause command changes are more frequent and aggressive.

III. CALIBRATION OF THE POSE ESTIMATION SYSTEM

This section explains how parameters required for poseestimation can be calibrated online using integrated per-

213 214 215 216 217 218 219 220 2210

1

2

3

4

5

time constant = 0.81, delay = 0.16 sec

time (s)

rad/s

left command

right command

left encoder

right encoder

left model

right model

Fig. 2. Left and right wheel angular velocities vs. time as the LandTamerdecelerates. The speeds predicted by the calibrated model closely match theactual (encoder) speeds.

225 230 235 240 245 250 255

0

1

2

3

4

5

time constant = 1.40, delay = 0.04 sec

time (s)

rad/s

Fig. 3. Wheel velocities vs. time as the LandTamer is teleoperated. Seelegend in Fig. 2

0 50 100 150 200 2500

0.5

1

1.5

time (s)

tim

e c

onsta

nt (s

)

ID

DE

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

tim

e d

ela

y (

s)

time (s)

Fig. 4. Estimates of powertrain parameters vs. time during the teleoperatedLandTamer test, using both the integrated dynamics (ID) and differentialequation (DE) online calibration approaches. The DE approach fails toidentify the time delay which affects the time constant estimate.

turbative dynamics (IPD). To demonstrate the process, wecalibrate the position of the IMU, steer angle sensors, and thewheel radius on the Zoe rover. Zoe previously surveyed thedistribution of biological life in Chile’s Atacama desert [13].Zoe has four independently driven wheels and two passivelyarticulated axles (see Fig. 5).

A. Odometry Equations

When performing dead reckoning in 3D (assuming a zyxEuler angle convention), the pose of the vehicle ρ in aground-fixed frame is predicted by integrating the following

(a) (b)

Fig. 5. (a) The Zoe rover in the Atacama desert. (b) A diagram of therover. The (W)orld, (R)obot, and (I)MU coordinate systems are shown inred.

kinematic differential equation:

ρ = f(ρ, u)xyθ

=

cθcβ cθsβsγ − sθcγ 0sθcβ sθsβsγ + cθcγ 0

0 0 cγcβ

VxVyVθ

(9)

c = cos(), s = sin(), γ = roll, β = pitch, θ = yaw

Where u = [Vx Vy Vθ]T is the velocity of the vehicle in the

body frame, and is computed as follows:

u(p) = H+v vVxVy

Vθ

=

12 0 1

2 00 1

2 0 12

0 1L 0 − 1

L

rw2

(ω1 + ω2)cψf(ω1 + ω2)sψf(ω3 + ω4)cψr(ω3 + ω4)sψr

(10)

In (10), L is the vehicle length, rw the wheel radius,and ω1 to ω4 are wheel angular velocities (see Fig. 5(b)).The front and rear steer angles (ψf , ψr) are sensed usingpotentiometers; measured steer angle is a linear function ofthe potentiometer output voltage V :

ψf = mfVf + bf , ψr = mrVr + br (11)

The pose of the robot at the end of the path segment ispredicted by integrating the kinematic differential equation(9) as follows:

ρ(t) = ρ(t0) +

∫ t

t0

f(ρ(τ), u(τ, p)

)dτ (12)

B. IMU Pose EquationsBy default, the GPS/INS unit on Zoe returns the pose

of the IMU with respect to the world frame (e.g. UTMcoordinates). In order to accurately estimate the pose of therobot with respect to the world frame we must know therelative pose of the IMU/robot frames.

For a path segment from t0 to t the predicted final pose ofthe IMU is a composition of poses (calculated by multiplyingthe corresponding homogenous transforms):

ρWIf

= ρWIi

∗ ρIiRi

∗ ρRiRf

∗ ρRfIf

ρWIf

= ρWIi

∗ (ρRI

)−1 ∗ ρRiRf

∗ ρRI

(13)

In (13), the lowercase subscripts i and f denote initial andfinal poses (i.e. at the beginning and end of a path segment).ρRI

is the current estimate of the pose of the IMU with respectto the robot frame. ρW

Iiis the GPS/INS measurement of the

initial pose of the IMU with respect to the world frame. ρRiRf

is the final pose of the robot with respect to the initial robotframe, and is computed using the odometry integral (12).

C. Online Identification of Pose Estimation Parameters

This section explains how the parameters required for poseestimation are calibrated online using an extended Kalmanfilter. Just as in the section on powertrain model identifi-cation, we identify all the necessary variables to perform ameasurement update.

The state vector contains the parameters we wish toestimate:

x =

[ρRI

podom

], p

odom= [mf bf mr br rw]T (14)

podom

is a vector of odometry parameters. The measurementz is the final pose of the IMU in the world frame ρW

If, as

measured by the GPS/INS. The predicted measurement h(x)is the predicted final pose of the IMU computed using (13)

The measurement uncertainty R depends on the accuracyof the pose measurement device. The measurement JacobianH can be considered in two parts:

H =

[∂ρW

If

∂ρRI

(∂ρW

If

∂ρRiRf

∂ρRi

Rf

∂podom

)](15)

∂ρWIf/∂ρR

Iand ∂ρW

If/∂ρRi

Rfare Jacobians of the pose

composition equation (13) with respect to intermediate poses;they can be solved for in closed form using symbolicmathematics software. ∂ρRi

Rf/∂p

odomis the Jacobian of the

odometry integral (12) with respect to the steer angle andwheel radius parameters.

The linearized error dynamics of the system help tocompute this Jacobian. The following convolution integralrelates input velocity perturbations to a change in terminalpose:

δρ(t) = Φ(t, t0)δρ(t0) +

∫ t

t0

Γ(t, τ)δu(τ,∆podom

)dτ (16)

In short, (16) attributes errors in our terminal pose predictionto errors in the initial pose measurement and velocity errorsδu, caused by errors in our odometry parameter estimates.Φ and Γ are the transition and input transition matrices:

Φ(t, τ) =

1 0 −(y(t) − y(τ))0 1 x(t) − x(τ)0 0 1

(17)

Γ(t, τ) = Φ(t, τ)∂f

∂u(18)

The transition matrix Φ(t, τ) accounts for the nonlineareffect that heading errors at time τ have on (x,y) position atthe future time t. Due to space limitations the derivation of

(17) and (18) is not included here, but can be found in [14].Applying (16), the rightmost term in H is:

∂ρRiRf

∂podom

=

∫ t

t0

Γ(t, τ)Up dτ (19)

where Up is the Jacobian of u in (10) with respect to podom

.The Leibniz rule is used to move the derivative inside theintegral.

D. Experimental Results

Experimental data was collected while driving the Zoerover by joystick at Flagstaff hill in Pittsburgh, Pennsylva-nia. Zoe is equipped with a high-end differential GPS/INSunit that produced independent pose measurements (withoutodometry aiding). Note however that GPS is not necessary;ground truth need only be accurate for relative pose mea-surements a few seconds apart so other low-cost techniquescould work such as visual registration of a few carefullysurveyed landmarks.

In a short traverse of just 163 m, the online IPD filter pro-duced accurate values for IMU position offsets and odometryparameters. The final parameter estimates are shown in TableII along with manually calibrated values. In the “Manual”column the IMU offsets and wheel radius were measuredwith a tape measure, and the steer angle potentiometer pa-rameters were calibrated manually using precision-machinedangle set blocks.

TABLE IICALIBRATED PARAMETER VALUES

Manual IPDxRI (cm) 38.0 46.02yRI (cm) -15.0 -19.86mf (rad/V) -0.1128 -0.1192bf (rad) -0.0054 0.0019mr (rad/V) -0.1080 -0.1181br (rad) -0.0105 0.0021rw (cm) 32.5 31.62

Although the IPD parameter values may seem to differlittle from the manual values, they produce a significantimprovement in pose prediction accuracy. Fig. 6 showsscatter plots of along-track and cross-track pose predictionerror for 6 second path segments. Notice the mean predictionerror using the IPD parameter estimates is reduced from 21cm to near zero, and the standard deviations of error arereduced by 20-40%. Quantitative results including headingerror are shown in Table III. The improvement is even morepronounced when integrating the entire predicted path (seeFig. 7). Note that, integrated odometry pose predictions usedIMU attitude measurements (γ, β) but not heading (θ).

Fig. 8 compares IPD performance to the classical (DE)approach which uses the differential equation (9) directly. Inthe classical approach, pose measurements are numericallydifferentiated to obtain velocity measurements:

ρ[k] =ρ[k + 1] − ρ[k − 1]

t[k + 1] − t[k + 1](20)

−0.4 −0.2 0 0.2 0.4 0.6

−0.2

−0.1

0

0.1

0.2

0.3

Pose prediction error − manual IMU offset & odometry cal.

cross track error (m)

alo

ng tra

ck e

rror

(m)

(a)

−0.4 −0.2 0 0.2 0.4 0.6

−0.2

−0.1

0

0.1

0.2

0.3

Pose prediction error − online IPD calibration

cross track error (m)

alo

ng tra

ck e

rror

(m)

(b)

Fig. 6. Scatter plots of pose prediction error for overlapping 6 secondpath segments. (a) using manual calibration parameter estimates. (b) usingonline IPD estimates, computed using 20-fold cross-validation.

TABLE IIIODOMETRY POSE PREDICTION ERROR1

Manual Online IPD2

Axis µ σ µ σ

Along-track (cm) 2.23 5.23 -0.28 4.12Cross-track (cm) -20.91 6.15 0.13 3.96

Heading (deg) -1.48 0.85 -0.01 0.491 for 6 second path segments2 computed using 20-fold cross-validation

GPS/INS

online IPD cal.

manual cal.

scale, 25 m

Fig. 7. GPS/INS path of the Zoe rover at Flagstaff hill. The integratedwheel odometry path using the online IPD parameter estimates is muchmore accurate than wheel odometry using manual calibration values.

0 1 2 30

0.05

0.1

0.15

0.2

0.25

time between meas (s)

mMean distance error

Manual

DE

IPD

(a)

0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

time between meas (s)

rad

Mean heading error

Manual

DE

IPD

(b)

Fig. 8. Comparison of IPD and DE (differential equation) approaches. (a)Mean (x,y) Euclidean distance error of pose predictions vs. the time betweenpose measurements (t-t0) during training. (b) Mean of the absolute valueof heading errors. Pose prediction errors are computed for 6 second pathsegments. When using the classical DE approach, pose prediction errorsincrease with the delay between measurements.

Accuracy of the numerical derivatives and performance ofthe DE approach degrade as the delay between pose mea-surments increases. Pose prediction errors in Fig. 8 arecomputed for 6 second path segments. This explains thespike in error in Fig. 8(a) when calibrating to very short pathsegments; IPD calibration results are best when the model istrained and evaluated using paths of comparable length.

In this test, wheel slip was kept to a minimum by Zoe’spassive steering design, high traction with the grassy terrain,and a speed limit of 1 m/s. Minimizing wheel slip isimportant when calibrating odometry parameters because ofobservability issues. For example, wheel radius error cannot be distinguished from longitudinal wheel slip that isproportional to forward velocity Vx. Intrinsic parametersmay also be indistinguishable; for example, θRI can notbe concurrently calibrated with bf and br. Parameters areindistinguishable if they produce linearly dependent columnsin the measurement Jacobian, H .

IV. CONCLUSIONS

This paper demonstrated how integrated dynamics (ID)and integrated perturbative dynamics (IPD) can be usedto automate the tedious but essential aspects of vehiclecalibration. The ID/IPD approach has advantages over clas-sically derived calibration techniques which use the systemdifferential equation directly. Specifically, ID/IPD enablesdirect use of state measurements instead of their numericalderivatives which can be extremely noisy. We used ID/IPDin combination with an extended Kalman filter to identifyintrinsic parameters of the vehicle powertrain and poseestimation system online.

Using integrated dynamics, we presented a method toidentify both the time constant and delay of a first-orderpowertrain model, without requiring double differentiationand bandpass filtering of encoder ticks. In addition, usingan online IPD algorithm we simultaneously calibrated IMUposition offsets and odometry parameters (steer angle sen-sors, wheel radius) on the Zoe rover in one short traverse.Pose predictions using the IPD approach were significantlymore accurate than using the manual or classical (differential

equation) calibration approaches, especially when delaysbetween pose measurements were large.

In conclusion, ID/IPD is an effective tool for the identi-fication of intrinsic robot parameters, especially when onlylow-frequency state measurements are available. In futurework we will apply active learning techniques (e.g. [15]) tocommand trajectories that will most reduce the uncertaintyof the parameter estimates.

ACKNOWLEDGMENT

This research was made with U.S. Government supportunder and awarded by the Army Research Office (W911NF-09-1-0557) and by the DoD, Air Force Office of ScientificResearch, National Defense Science and Engineering Gradu-ate (NDSEG) Fellowship, 32 CFR 168a. The authors wouldalso like to thank David Wettergreen and Dominic Jonak formaking Zoe available for testing.

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