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4WD Skid-Steer Trajectory Control of a Rover with Spring-Based Suspension Analysis

Direct and Inverse Kinematic Parameters Solution

Edgar Mart? ?nez-Garc? and Rafael Torres-C?rdoba

?a o

Robotics Laboratory Institute of Engineering and Technology Universidad Aut?noma de Ciudad Ju?rez, M?xico o a e

Abstract. This manuscript provides a solution to the problem of fourwheel drive (4WD) kinematics and dynamics for trajectory control of an in-wheel motors rover. The rover is a platform built up in our laboratory, featured by its damper devices. The rover wheels’ contact point have dynamic positions as e?ects of spring-based suspension devices damper. The tracking control is a four-wheel drive skid-steering (4WDSS) system, and we propose a motion control fundamentally de?ning a dynamic turning z-axis, which moves within the area of the four wheels’ contact point. We provide a general solution for this mechanical design since the wheels’ contact point displacement directly impacts the rover angular velocity. Furthermore, we introduce a model for inertial localization based on an arrangement of two accelerometers to de?ne the rover position within a global inertial frame.

1

Introduction

This manuscript provides a formulation in continuous time to calculate inverse and direct kinematics parameters for trajectory control of a four-wheel drive (4WD) mobile robot with in-wheel motors and independent suspensions (?g.1) to exploit the skid-steering ability. One the problems stated here is de?ned as how to describe trajectory control with asynchronous wheels drive based on skidsteering (4WDSS) [7][8][5], since robot angular acceleration ωt works di?erent with respect to (w.r.t.) other traditional synchronous drive control systems. The angular acceleration will depend on the wheels position overtime as the damper device cause e?ects in L1 and L2 (?g.1). The authors present three di?erent general solutions for the angular speed: a) under no damper constraints in totally plain terrains; b) under spring-mass e?ects in plain ?oors; and c) under uneven terrains where the normal force is considered. Furthermore, the models presented include dead-reckoning analysis with kinematic and dynamic parameters solutions, which are used to characterize real trajectory control. Many trajectory control research reported in the related work present approaches to control yaw using di?erent sensing modalities with fundamentals in the rover kinematics of non-holonomic systems [7], [1], [4], skid control properties [8] [10], and direct

H. Liu et al. (Eds.): ICIRA 2010, Part I, LNAI 6424, pp. 453–464, 2010. c Springer-Verlag Berlin Heidelberg 2010

454

E. Mart? ?nez-Garc? and R. Torres-C?rdoba ?a o

control over the e?ectors [3]. Other works, include to this problem the ability to control position and angle steer to achieve tracking control [2], [6], or preserving stability [5], and in some cases obstacle avoidance is also complemented [1]. Our approach combines simple low cost devices to accomplish trajectory tracking and inertial localization for all terrains. The section 2 describes the mechanical design and kinematic parameters of the 4WD rover. Section 3 details the rover angular velocity model under di?erent kinematic conditions. The section 4 describes in detail how the suspension in?uences the wheels’ contact point locations, and how such model is integrated with the angular velocity model. The section 5 describes the trajectory control model and the dynamic rotation axis w.r.t. the instantaneous center of rotation. Section 6 provides the parameter solution for the 4WDSS inverse kinematics. Section 7 is devoted to the inertial unit model for inertial localization, describing the results on experimental implementations. Finally, section 8 provides the paper conclusions.

2

Robotic Platform Kinematic Con?guration

According to ?g.1 a four-wheel drive skid-steer (4WDSS) robotic platform with ˙ i independent wheels control speeds ?i is analysed. The distance lt between wheels t and the geometric center is variable on time as the robot moves with angles αi . The real distance between contact points on one robot’s side is de?ned by t Li + Li+1 . The rover’s angular velocity and bearing direction are de?ned by ωt t t and θt respectively. The angle βt is non-stationary and represents a kinematic restriction featured by the angle between the rotational wheel axis, and the line which cuts the rover center of mass and the wheel contact point [9]. The rover inertial system is de?ned locally by the X along its longitudinal axis, and Y in its transversal axis. There exist a mechanical arm of length d for each wheel, with two joints. On one extreme, the suspension is ?xed around the middle rover side with a rotative

X

t

3

3 t

t

t

1

t

1

t

r l3t

Y

3 4

Lft

1 t t 2 t

Ltf

l1t

t

chasis

L1

t

ms d1

dv/dt

y

l4

4 t

t

Lrt

l 2t

L2t

d

2

d /dt

4

t

1

t

fn floor w/4

t

Fig. 1. Left: Top view of the rover-like mobile robot with its kinematic con?guration. Right: Kinematic con?guration of the spring-based suspension of each wheel.

4WD Skid-Steer Trajectory Control of a Rover

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mechanical device that allows free angular motion. The another joint is a springbased damper device located around the middle of the arm tied vertically to the f,b chassis plate (as depicted in ?g.2-right). The angle γt represents the angular position of each suspension arm, measured with a linear transducer implemented in the rotative arm device.

3

Angular Speed Models

Since the rover has independent wheels speed control, its instantaneous velocity model is de?ned by the averaged wheels’s speeds as given by the equation (1), vt = r 4

4

?i ˙t

i=1

(1)

Where r represents the nominal wheels radius (assuming in this manuscript all wheels have the same r magnitude), and ?i is each of the rotational wheels’ ˙t speed. Furthermore, the rover’s di?erential linear velocity vt is formulated as in ? the expression (2), vt = r(?1 + ?2 ? ?3 ? ?4 ) ? ˙t ˙t ˙t ˙t (2)

where left-sided wheel rotational speeds are denoted to have minus sign w.r.t. the right-sided wheels. Therefore, the general model for the robot angular velocity is de?ned respect to its center of mass, ideally in its geometrical center by ωt = v cos(θt ) ? Lf t = r cos(θt ) Lf t (?1 + ?2 ? ?3 ? ?4 ) ˙t ˙t ˙t ˙t (3)

θt is the bearing robot direction at continuous time t, and the reference point of rotation is taken from the middle front side of the robot namely Lf . Also, t representing the contact point projection over the front side longitudinal vehicle axis. Now, we can easily de?ne the control vector by ? 1? ?t ˙ r r r r ??2 ? vt ˙t ? 4 4 4 4 (4) ut = = r cos(θt ) r cos(θt ) r cos(θt ) r cos(θt ) ? 3 ? ??t ? ? ωt ˙ Lf Lf Lf Lf t t t t ?4 ˙t where ut = (v, ω)T is control vector by which the robot is moved toward a desired local goal destination in terms of direct kinematics [13,11]. While next expression de?nes velocity state control vector but in terms of inverse kinematics, ? ? ? x ˙ cos θt ˙t = ?y ? = ? sin θt ˙ ξ ˙ 0 θ ? 0 0? ut 1

(5)

456

E. Mart? ?nez-Garc? and R. Torres-C?rdoba ?a o

3.1

Case for No Spring-Mass Model

The following is a set of equations for modelling ωt with kinematics motion only. For this case the suspension has no energy release because the rover moves around a totally man-made ?at ?oor. Firstly, let us de?ne a general solution for αi by (see ?g.1), t sin(αi ) = t Lf t ; i lt

i Lf = d cos(γt ) t

(6)

It follows a formulation of linear equations to project Li . By knowing a model t for Lf , wheels position can be inferred to calculate the robot angular velocity. i Lf is substituted to de?ne a ?rst equation, t

i i lt sin(αi ) = d cos(γt ); t i i lt sin(αi ) ? d cos(γt ) = 0 t

(7)

likewise, second equation is de?ned W i tan(αi ) = d cos(γt ); t 2 W i tan(αi ) ? d cos(γt ) = 0 t 2 (8)

and the third equation is also de?ned by W/2 i = d cos(γt ); cos(αi ) t W i ? d cos(γt ) = 0 2 cos(αi ) t (9)

For (9) we de?ne an equation in terms of αi , t αi = t W i 2 cos(γt ) (10)

i Similarly, for calculating lt using (7) and (8), and algebraically arranging, i lt = sin(αi ) = t

W tan(αi ); t 2

i lt =

W tan(αi ) t 2 sin(αi ) t

(11)

i with lt dropped o?, let us substitute functional form of αi t W W tan(arccos( 2d cos(αi ) ))

t

i lt =

W 2 sin(arccos( 2d cos(αi ) )

t

(12)

Thus, the following model is a limit case where no spring-damper is considered, including a de?nition for κ, ωt = 2r cos(θt ) κ

4

?i ; ˙t

i=1

κ=

W tan(αi ) i t α 2 sin(αi ) t t

(13)

4WD Skid-Steer Trajectory Control of a Rover

457

4

Suspension Motion Analysis

The general spring-mass formulation is de?ned by (14) according to the geometric suspension design setup as a vertically aligned spring-based device. The model regards two constants in the equation, the spring restitution coe?cient κ1 , and the damper coe?cient κ2 . Ft = ?κ1 Δy ? κ2 y n + ˙ w 4 (14)

Where, Ft is the force required to exert spring linear motion at time t, its motion yields a displacement vertically namely Δy with a spring friction constant κ1 . The spring-mass damper term is κ2 y n , with n = 1 which is a linearisation ˙ parameter. This term is associated to the spring instant linear velocity. The vehicle weight w is divided by four-wheel contact points, so that the right-sided term of the equation (14) is denoted by w/4. The spring device displacement is de?ned by (15)

i Δy = d1 cos(γt )

(15)

Thus, in terms of spring-mass and acceleration,

i ms yt = ?κ1 d1 cos(γt ) ? κ2 y n + ¨ ˙

w 4

(16)

Where ms is the spring mass, and at is the acceleration motion yielded by the spring dampering e?ects. Algebraically arranging , let us substitute (16) into the i equation (6) in terms of cos(γt ), thus

i cos(γt )

=

¨ ˙ ms y t + κ 2 y + ?κ1 d1

w 4

(17)

4.1

Condition for Spring Damper

The condition for motion over totally even terrains or structured ?oors, the normal force is declared as fn = 0 because of consideration of no spring motion is regarded. Therefore, substituting in terms of Lt with fn = 0, Lf = t d w (ms y ? κ2 y ? ) ¨ ˙ ?κ1 d1 4 (18)

Hence, the vehicle angular rotation speed constrained by its dynamic wheels contact point is presented by next equation, ωt = ?κ1 d1 r cos(θt ) ˙2 ˙3 ˙4 ˙1 w (?t + ?t ? ?t ? ?t ) d(ms y ? κ2 y ? ) ¨ ˙ 4 (19)

458

E. Mart? ?nez-Garc? and R. Torres-C?rdoba ?a o

5

Trajectory Control Model

For trajectory control we combine the instantaneous center of rotation (ICR), from which there exist a radius R to the robot rotation point coordinate (xR , yR )T . Let us de?ne a general assumption, r i ?t ˙i v = (20) ω 4ωt The robot angular velocity ωt is now a known model that exert motion control e?ects over the robotic platform and can be de?ned for di?erent cases: R= 1. Rigid or no suspension (?xed 4WD) ωt = ˙t ˙t ˙t (?1 + ?2 ? ?3 ? ?4 ) cos(θt ) ˙t Lf t

2. Suspension with no damper e?ects (totally even ?oor and smooth turns) ωt = ˙t ˙t ˙t ˙t 4 sin(αt )(?1 + ?2 ? ?3 ? ?4 ) cos(θt ) W tan(αi )αi t t ˙t ˙t ˙t ˙t ?κ1 d1 r(?1 + ?2 ? ?3 ? ?4 ) cos(θ) d(ms y ? κ2 y ? w/4) ¨ ˙

3. Suspension with spring-mass e?ects ωt =

However, the third case could be chosen as the most general case substituting the ?rst two cases (if required). Our proposed strategy establishes that the robot trajectory is tracked by the robot placing over the path a robot rotation axis which is located at (xR , yR )T . The rotation axis is dynamically positioned within the area of the four wheel contact points, depending on the inertial robot behaviour. Thus, this dynamic e?ect is fundamental in the 4WDSS nature of the robot (see also [5]). The rotation axis is then formulated within the robot’s inertial ?xed frame for XY axis, xR = W (?1 ? ?2 ? ?3 + ?4 ); ˙ ˙t ˙t ˙t vmax t yR = vt L (??1 + ?2 ? ?3 + ?4 ) ˙t ˙t ˙t ˙t v max (21)

The rotation axis coordinates will move nearly around the robot centroid according to its motion direction. Thus, the ICR at coordinates < xc , yc > is mathematically de?ned by, xc yc = (xt + xR ) ? R sin(θt + ωt Δt ) (yt + yR ) + R cos(θt + ωt Δt ) (22)

By deducing an ideal robot position recursively w.r.t. the ICR model [13], xt yt = xt?1 yt?1 + xc ? xR ? R sin(θt + ωt Δt ) yc + yR + R cos(θt + ωt Δt ) (23)

4WD Skid-Steer Trajectory Control of a Rover

459

v3

V1 v4 <xR,yR> <x,y> V2 R <xc,yc>

Fig. 2. Left: 4WD skid-steer trajectory control and turning axis; Right: A photo of the rover-like mobile robot

6

Inverse Kinematic Parameters Solution

By means of inverse kinematics analysis wheels control parameters are solved, and trajectory control and stability are better de?ned [6]. The wheels kinematic parameters are featuring low level e?ector commands[10], hence robot location and robot velocities can be established for navigation control. Next is a set of linear equations de?ned in terms of direct kinematics. 4vt r d(ms y ? κ2 y ? w/4) ¨ ˙ 1 2 3 4 ?t + ?t ? ?t ? ?t = ωt ˙ ˙ ˙ ˙ ?κ1 d1 r cos(θt ) xR v max ?1 ? ?2 ? ?3 + ?4 = ˙t ˙t ˙t ˙t W yR v max 1 2 3 4 ??t + ?t ? ?t + ?t = ˙ ˙ ˙ ˙ L ˙t ˙t ˙t ?1 + ?2 + ?3 + ?4 = ˙t

(24)

By solving the set of linear equations by Gauss-Jordan (or any other factorization method), we obtain the wheels velocity vector control Ωt = (?1 , ?2 , ?3 , ?4 )T ˙t ˙t ˙t ˙t which is the general solution for inverse kinematics, vt λ1 + ωt λ2 + xR λ3 ? yr λ4 4λ1 λ2 λ3 λ4 vt λ1 + ωt λ2 ? xR λ3 + yr λ4 ?2 = ˙t 4λ1 λ2 λ3 λ4 ?vt λ1 ? ωt λ2 ? xR λ3 ? yr λ4 3 ?t = ˙ 4λ1 λ2 λ3 λ4 vt λ1 ? ωt λ2 + xR λ3 + yr λ4 ?4 = ˙t 4λ1 λ2 λ3 λ4

?1 = ˙t

(25)

460

E. Mart? ?nez-Garc? and R. Torres-C?rdoba ?a o

And de?ning the factors λi the next equation is described as, λ1 = 1 ; r λ2 = ?κ1 d1 r cos(θ1 ) ; d(ms y ? κ2 y ? w/4) ¨ ˙ λ3 = vmax ; W λ4 = vmax L

The purpose of this formulation is to calculate the wheels angular speed vector ˙t ˙t ˙t ˙t Ωt = (?1 , ?2 , ?3 , ?4 )T , which is essential to solve for the control vector ut in order to reach skid-steering trajectory control. Thus, by solving the linear equation system of (24) results are depicted in ?g.4 that shows some simulated skid-steer based control trajectories (and empirically demonstrated in laboratory) similarly reported in [5].

Y X

Y X

Z

Z

Fig. 3. Left: Di?erent trajectories yielded by changing robot width W ; Middle and Right: Robot trajectory controlled by parametrizing Ωt to shift the Z-turning axis < xR , yR > denoted by the bold points

7

Inertial Localization

The main role of robot position regards as a parameter for the instantaneous center of rotation model in the trajectory control. A home made inertial module compounded of two accelerometers to measure displacement in a global framework was developed. Deploying inertial sensors for 4WDSS trajectory control works on a dead-reckoning modality, and wheel encoders [12] are not suitable for skid-steering techniques, where rough and uneven terrains are faced. The inertial module was built up with a couple of accelerometers, geometrically arranged at the front and rear side the robot, along the longitudinal robot axis, since only one accelerometer is able to measure respect to its ?xed frame. 7.1 Global Frame Measuring Strategy

The angular acceleration is measured as a means to determine the total acceleration of a chassis point (see ?g.4-left). The angular acceleration causes a tangential acceleration which is a component perpendicular to the position vector of such point, where the acceleration is being calculated with the rotation center of the body w.r.t. the origin. The orientation of the tangential acceleration bears toward a direction of the α and perpendicular to a position r, from the rotation center G of the robot, de?ned by next equation (a de?nition is described in [14]),

4WD Skid-Steer Trajectory Control of a Rover

461

aAG = ?α × rA ? ω 2 rA ;

aBG = ?α × r B ? ω 2 r B

(26)

Where aAG is the relative acceleration of point A (the front side) with respect to G (geometric center). The relation ω 2 r is also de?ned by ω 2 r = ω × ω × ×r. aBG is the relative acceleration of point B (the rear side) with respect to G, r A is the position vector of point A with respect to G. rB is the position vector of point B with respect to G, α is the angular acceleration vector of the robot, which is normal to the reference plane. The total acceleration of a point is equal to the acceleration of the rotation center G, plus the relative acceleration of the point with respect to G. Thus, for points A and B the following model can be stated, aA = aG + aAG ; aB = aG + aBG (27)

Where aG is the acceleration of the rotation center of the object. If total accelerations of A and B are subtracted, the acceleration of G is taken out, so this acceleration di?erence is independent of the chassis acceleration. aB ? aA = aBG ? aAG If equations (27) are substituted in (28) the result is as follows, aB ? aA = rB × α + ω 2 r B ? (rA × α + ω 2 r A ) aB ? aA = (r B ? r A ) × α + ω 2 (r B ? r A ) After simpli?cation it leads to: aAB = r AB × α + ω 2 r AB (31) (29) (28)

(30)

Being aAB the di?erence of accelerations of A and B. By de?nition, rAB × α and ω 2 r AB are orthogonal, so the following expression is valid. aAB = ( rAB α )2 + ( rAB ω 2 )2 (32)

If we solve for α , the following equation turns out, α= α = aAB rAB

2

? ω4

(33)

With this resulting equation is possible to calculate the angular acceleration of the moving object if the acceleration of two di?erent geometric points are known. This formula includes the angular speed, which depends on the angular acceleration itself. As for implementation, such angular acceleration needs to be calculated iteratively using sensor readings. The following is a recursive

462

E. Mart? ?nez-Garc? and R. Torres-C?rdoba ?a o

algorithm to calculate the angular acceleration, the angular speed, and the orientation angle of the robot, (axB ? axA )2 + (ayB ? ayA )2 l ωt = ωt?1 +

t 2 4 ? ωt?1

αt =

(34)

αt dt;

θt = θt?1 +

t

ωt dt

(35)

Where αt is the current angular acceleration of the robot. axA , axB , ayA and ayB are the acceleration values obtained from both two-axis accelerometers. l is the distance between the position of the accelerometers on the robot. ωt is the current angular speed of the robot. Δt is the time interval between each sensor reading. And θt is the current angle of the robot. 7.2 Absolute Position

With the orientation angle being known, the acceleration and therefore the position of the system can be obtained according to an inertial reference frame, carried out by calculating the rotation of the accelerometers local frames, and integrating the result. The inertial coordinates of the front side accelerometer is given by (axA cosθ ? ayA sinθ)dt2 ;

t t

xAI =

y AI =

t t

(axA sinθ + ayA cosθ)dt2 (36)

while the inertial coordinates of the rear side accelerometer are, xBI =

t t

(axB cosθ ? ayB sinθ)dt2 ;

yBI =

t t

(axB sinθ + ayB cosθ)dt2 (37)

xI and yI are the location of each accelerometer in an inertial frame, and where xt and yt are the current position of the robot, θt is the current orientation angle, xt?1 , yt?1 , θt?1 are the last pose, vt and ωt are the current linear and angular speeds, respectively, and Δt is the time interval between consecutive measures. 7.3 Experimental Analysis

The mobile robot used in this research work is a home made platform (see ?g.2right). The rover can be con?gured in 4WDD (di?erential drive) or 4WDSS (asynchronous speeds) control modality. Experimentally, the robot was setup in tele-operation mode collect massive data, throughout an ideal curved trajectory (?g.4-right), and computing the real position with the inertial localization formulation in real-time. The proposed two-accelerometer algorithm yielded position errors which increased as the robot navigates. Several main disturbance causes were experimentally identi?ed that yielded error positions. Accelerometers sensitivity took an important roll despite of the home-made inertial unit

4WD Skid-Steer Trajectory Control of a Rover

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Fig. 4. Left: Array of two accelerometers instrumenting the 4WDSS rover. Right: Trajectory control in experimental results for Ωt and inertial localization under skidsteering motion control.

system capability to combine information. A couple of two low-cost accelerometers were the unite core. We found out that more accurate devices are required to reduce errors in the data fusion formulation. In earlier experiments, systematic causes were also detected to be a problem such as wheels mechanical shaking, and a suitable wheels sliding (lateral and longitudinal) formulation has not been integrated in this work yet. An optimal state estimation nonlinear ?lter for position measurement will be a matter of integration for the proposed model in the future. Likewise, a means based on GPS will be developed to reduce the accumulated error overtime within navigation segments where no speed changes are detected (although GPS was used, results are not depicted due to space in this manuscript).

8

Conclusions

A general solution to inverse and direct kinematics for trajectory control of a four-wheeled mobile robot was proposed. The mathematical formulation provided a model to explicitly control the robot’s angular velocity in terms of its dynamic contact points geometry and the wheels rotational rate, including inwheel spring-based damper device. For the inertial localization, several main disturbance causes were experimentally identi?ed that yielded error positions, such as accelerometers accuracy, the systematic causes such as wheels mechanical shaking. A suitable slipping formulation will be integrated, as well as position state estimation ?ltering. The paper provides a deterministic approach for three di?erent damper conditions, establishing general solutions of the yaw speed. Finally, the presented framework has experimentally resulted feasible to be implemented in robotic platforms and trajectory control was stable enough for many navigational tasks.

464

E. Mart? ?nez-Garc? and R. Torres-C?rdoba ?a o

Acknowledgment

The authors would like to thank the partial support provided by PROMEP under project No. DGPDI/SPROMEP/2-9/075.

References

1. Ki, K., Wang, S., Di, H., Li, H.: Trajectory tracking control wheeled mobile robots based on integrated intelligent steering. In: IEEE ICEMI 2009, pp. 3-955–3-960 (2009) 2. Solea, R., Filipescu, A., Nunes, U.: Sliding-mode control for trajectory-tracking of a wheeled mobile robot in presence of uncertainties. In: Proc. of the 7th Asian Control Conference, Hong Kong, China, August 27-29, pp. 1701–1706 (2009) 3. Zhao, Y., Zhang, Y., Zhao, Y.: Stability control system for four-in-wheel-motor drive electric vehicle. In: IEEE 6th Intl. Conf. on Fuzzy Systems and Knowledge Discovery, pp. 171–175 (2009) 4. Silva-Ortigoza, R., Silva-Ortigoza, G., Hernandez-Guzman, V., Barrientos-Sotelo, R., Albarr?n-Jim?nez, Silva-Garcia, V.: Trajectory tracking in a mobile robot witha e out using velocity measurement for control of wheels. IEEE Latin America Transaction 6(7), 598–607 (2008) 5. Kozlowski, K., Pazderski, D.: Practical stabilization of a skid-steering mobile robot - A kinematic-based approach. In: IEEE Intl. Conf. on Mechatronics, pp. 519–524 (2006) 6. Caracciolo, L., De Luca, A., Iannitti, S.: Trajectory Tracking Control of a FourWheel Di?erentially Driven Mobile Robot. In: Proc. IEEE Intl. Conf. on Robotics and Automation, Detroit, Michigan, USA, pp. 2632–2638 (May 1999) 7. Mandow, A., Mart? ?nez, J.L., Morales, J., Blanco, J.L., Garc? ?a-Cerezo, A., Gonz?lez, a J.: Experimental Kinematics for Wheeled Skid-Steer Mobile Robots. In: Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2007, San Diego, California, pp. 1222–1227 (2007) 8. Kozlowski, K.: Modelling and control of a 4-wheel skid-steering mobile robot. Int. J. Appl. Math Comput. Sci. 14(4), 477–496 (2004) 9. Campion, G., Baston, G., Andrea-Novel, B.D.: Structural properties and classi?cation of kinematic and dynamic models of wheeled mobile robots. IEEE Transaction on Robotics and Automation 12(1) (February 1996) 10. Wang, D., Low, C.B.: Modeling and analysis of skidding and slipping wheeled mobile robots: control design perspective. IEEE Transactions on Robotics 24(3), 676–687 (2008) 11. Macek, K., Thoma, K., Glatzel, R., Siewart, R.: Dynamics modeling and parameter identi?cation for autonomous vehicle navigation. In: Proc. of the 2007 IEEE/RSJ Intl. Conf. on Intelligent Robots and Systems, pp. 3321–3326 (2007) 12. Antonelli, G., Chiaverini, S.: Linear estimation of the odometric parameters for di?erential-drive mobile robots. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3287–3292 (2006) 13. Thrun, S., Burgard, W., Fox, D.: Probabilistic Robotics. MIT Press, Cambridge (2005) 14. Beer, F.P., Johnson, E.R., Eisenberg, E.R., Sarubbi, R.G.: Vector mechanics for engineers dynamics, ch. 15m, 5th edn., pp. 722–723. Mc Graw Hill, New York

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