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Backstepping based trajectory tracking control of a four wheeled mobile robot Umesh Kumar & Nagarajan Sukavanam Department of Mathematics, Indian Institute of Technology, Roorkee , India Corresponding author E-mail :(umeshdma@iitr.ernet.in) Abstract : For a four wheeled mobile robot a trajectory tracking concept is developed based on its kinematics A trajectory is a time–indexed path in the plane consisting of position and orientation The mobile robot is modeled as a non holonomic system subject to pure rolling , no slip constraints.To facilitate the controller design the kinematic equation can be converted into chained form using some change of co-ordinates.From the kinematic model of the robot a backstepping based tracking controller is derived Simulation results demonstrate such trajectory tracking strategy for the kinematics indeed gives rise to an effective methodology to follow the desired trajectory asymptotically Key words :wheeled mobile robot, chained form systems, nonholonomic systems, trajectory tracking Introduction A mobile robot is one of the well known system with nonholonomic constraints and there are many works on its tracking control Their objective are mostly the kinematics model For non holonomic systems such as mobile robots their kinematics constraints make time derivative of some configuration variables nonintegrable (Xiaoping,Y & Yamamoto,Y.,1996) Due to the appearance of the nonholonomic constraints the motion planning and the tracking control of mobile robots are difficult to be managed In the phase of motion planning (Wilson ,D.E.,and Luciano,E.C.,2002) a suitable trajectory is designed to connect the initial posture (i.e the position and orientation of the robot) and the final one such that no collisions with obstacles would occur and kinematics constraints are satisfied Once the feasible path is obtained the navigation and control process enters the tracking phase In this paper we restrict our attention to the kinematics of the nonholonomic systems such that every path can be followed efficiently In (Kanayama, Y ,kimura, Y ,Miyazaki,F & Noguchi,T.,1990) a linearization based tracking control scheme was introduced for a two degree of freedom mobile robot A similar idea was independently examined by Walsh et al in (Walsh, G., Tilbury, D , Sastry, S., Murray, R & Laumond, J.P.,1994).The idea of input-output linearization was further explored by (Oelen,W & Amerongen ,J.,1994) for a two degree of freedom mobile robot All these papers solve the local tracking problem for some classes of nonholonomic systems The class of nonholonomic system in chained form was introduced by (Murray, R M., & Sastry , S.S.,1993) and has been studied International Journal of Advanced Robotic Systems, Vol 5, No.4 (2008) ISSN 1729-8806,pp.403-410 as a bench mark example by several authors It is well known that many mechanical system with nonholonomic constraints can be locally or globally, converted to chained form under coordinate change and state feedback Interesting examples of such mechanical systems include tricycle-type mobile robots ,cars towing several trailers, the knife edge(Murray, R M., & Sastry , S.S.,1993), (Kolmanovsky ,I & McClamroch, N H ,1995) Trajectory planning algorithm for a four-wheel-steering (4WS) vehicle based on vehicle kinematics was introduced by (Danwei, W & Feng, Q.,2001).Trajectory tracking control of tri-wheeled mobile robots in skew chained form system was introduced by(Tsai, P.S.,Wang, L.S.,& Chang, F.R.,2006) In this paper we develop the kinematic model of a four wheel nonholonomic mobile robot subject to pure rolling and no side slipping condition and we propose a systematic control design procedure for chained form obtained from the kinematic model.The stepwise control design procedure used in this paper is based on the backstepping approach The problem to be solved here is the tracking of a desired reference trajectory for a four wheeled mobile robot moving on a horizontal plane and simulation results were presented to illustrate the approach Model of the mobile robot 2.1 Introduction In this section kinematic model of the four wheel nonholonomic mobile robot is presented In order to simplify the mathematical model of the four wheel nonholonomic wheeled mobile robot we assume that 403 International Journal of Advanced Robotic Systems, Vol.5, No.4 (2008) a The wheeled mobile robot is built from rigid mechanism b There is zero or one steering link per wheel c All steering axes are perpendicular to the surface of motion d The surface is a smooth plane e No slip occurs between the wheel and the floor 2.2 Instantaneous centre of rotation (ICR) It is an imaginary point around which a rigid body appears to be rotating momentarily (for an instant) when the body is rotating and translating For a four wheel mobile robot the instantaneous centre of rotation is the cross point of all axes of the wheels Its importance lies in the fact that wheel axes must intersect at a point if there is no slipping Fig.1:Four wheel mobile robot and co-ordinate frame matrices are × rather than × and are given by 2.3 Kinematic model of the four wheel mobile robot In this section the kinematic model is developed.A wheeled mobile robot is a wheeled vehicle which is capable of an autonomous motion (without external human driver) because it is equipped ,for its motion ,with motors that are driven by an embarked computer The four wheel mobile robot considered in this paper is shown in figure 1, its front wheels are steering wheels, and its rear wheels are driving wheels The distance between the front wheel axle and platform centre of gravity is c and distance between the rear wheel axle and platform centre of gravity is d and 2b is the wheel span The trajectory planning will be done for the platform centre of gravity Let the generalized co-ordinates be q = [ x0 , y0 ,φ0 ,φ ]T ,where (x , y ) 0 are the cos φ0 =  sin φ0  0 1T − sin φ0 cos φ0 x0  y0  ,  cos φ1 − sin φ1 c  =  sin φ1 cos φ1 b   0  2T cos φ2 =  sin φ2  − sin φ2 cos φ2 0 3T 1 − d  1 − d  = 0 b  , 04T = 0 −b  0  0  c −b  ,  cartesian coordinates of the centre of gravity of the mobile platform with respect to co-ordinate frame {U } The four wheels are located at p1 , p2 , p3 and p4 on the mobile platform respectively U 0T 2.4 Velocities pc is the centre of the mobile platform Six co-ordinate frames are defined for describing position and orientation of the mobile robot {1} is the frame fixed on wheel x1 - axis is choosen to be along the horizontal radial direction and y1 axis in the lateral direction Likewise {2},{3} and {4} are the frame defined for the wheel 2, and respectively {0} is the frame defined at point pc The orientation of the vehicle body is characterized by φ which is the angle from xU to x0 φ1 and φ2 are the front two steering angle and φ is the angle at which the whole platform changes the orientation due to steering angles φ1 and φ2 of the front two steering wheels With these notations we establish homogenious transformations describing one frame relative to another a bT denotes the homogenious transformation of frame {b} relative to frame {a} Because the motion of the mobile robot is restricted to 2dimentional plane homogenious transformation 404 Fig.2:Velocities of wheels With the help of homogenious transformation given above the velocities of point p1 , p2 , p3 and p4 can be easily computed The homogenious position vector of point p1 in Umesh kumar & Nagarajan sukavanam: Backstepping based trajectory tracking control of a four wheeled mobile robot frame {0} is condition that the wheels can not move in the lateral c  P1 = b  1  The same point is represented in frame {U } by direction (i.e y-component of U P1 = U0T P1  c cos φ0 − b sin φ0 + x0  = c sin φ0 + b cos φ0 + y0    − sin(φ0 + φ1 ) x&0 + b sin φ1 φ&0 + c cos φ1 φ&0 + cos(φ0 + φ1 ) y& = − sin(φ0 + φ2 ) x&0 − b sin φ2 φ&0 + c cos φ2 φ&0 + cos(φ0 + φ2 ) y& = − sin φ x& − d φ& + cos φ y& = 0  −c sin φ0 φ&0 − b cos φ0 φ&0 + x&0    & P1 =  c cos φ0 φ&0 − b sin φ0 φ&0 + y&      In order to derive the nonholonomic constraint equation of wheel The velocity of point p1 relative to frame 0 cos(φ0 + φ1 ) x&0 − b cosφ1 φ&0 + c sin φ1 φ&0 + sin(φ0 + φ1 ) y&0 = rθ&1 cos(φ0 + φ2 ) x&0 + b cosφ2 φ&0 + c sin φ2 φ&0 + sin(φ0 + φ2 ) y&0 = rθ&2 cosφ x& − b φ& + sin φ y& = rθ& 0 in frame {U } is then P&1 , P&2 , P&3 and P&4 are all zero).So finally we have seven constraints given by The velocity of point p1 relative to frame {U } expressed 0 cosφ0 x&0 + b φ&0 + sin φ0 y&0 = rθ&4 U {U } is expressed in frame {1} is θ are the angular displacement of the wheels Choosing the following as generalised co-ordinate vector q = [ x0 , y0 ,θ1 ,θ ,θ3 ,θ ,φ0 ,φ ]T The seven constraint can be written as A ( q ) q& = P&1 = ( U T1 ) −1 U P&1  cos φ0  =   sin φ0    −1 − sin φ0 cos φ0 x0  cos φ1 − sin φ1 c    y0   sin φ1 cos φ1 b   U P&1    0    cos(φ0 + φ1 ) x&0 − b cos φ1 φ&0 + c sin φ1 φ&0 + sin(φ0 + φ1 ) y&    =  − sin(φ0 + φ1 ) x&0 + b sin φ1 φ&0 + c cos φ1 φ&0 + cos(φ0 + φ1 ) y&      (1) where A is a × matrix given by A(q ) =  − s in ( φ + φ )   − s in ( φ + φ )  − s in φ  c o s (φ + φ )   c o s (φ + φ )  cos φ0   cos φ0  Similarly the velocity of point 2, 3, relative to frame {U } expressed in frame {2} , {3} and {4} as follows c o s (φ + φ ) c o s (φ + φ ) cosφ0 s in ( φ + φ ) s in ( φ + φ ) 0 −r 0 0 −r 0 0 0 0 0 s in φ s in φ 0 0 −r 0 −r c c o s φ + b s in φ 0  0 0  0 0  0   cos(φ0 + φ2 ) x&0 + b cos φ2 φ&0 + c sin φ2 φ&0 + sin(φ0 + φ2 ) y&    2& P2 =  − sin(φ0 + φ2 ) x&0 − b sin φ2 φ&0 + c cos φ2 φ&0 + cos(φ0 + φ2 ) y&0       cosφ0 x&0 − b φ&0 + sinφ0 y&0    = − sinφ0 x&0 − d φ&0 + cosφ0 y&0       cosφ0 x&0 + b φ&0 + sinφ0 y&0    4& P4 = − sinφ0 x&0 − d φ&0 + cosφ0 y&0      c c o s φ − b s in φ −d c s in φ − b c o s φ c s in φ + b c o s φ −b 3& P3 2.5 Constraint equations To develop the kinematic model of the wheeled mobile robot ,the ith wheel is considered as rotating with angular velocity θ& where θ& , i = 1, 2,3, denotes the angular i and θ1 ,θ , θ3 and Where r is the radius of each wheel i velocities for each wheel.For simplicity the thickness of the wheel is neglected and is assumed to touch the plane at Pi as illustrated in fig.2.Further it is assumed that during the motion the plane of each wheel remain vertical and the wheel rotates around its (horizontal) axle whose orientation with respect to the frame can be fixed or varying The first four constraints in which two constraint are identical are due to no slip condition i.e pure rolling and the other four constraints obtained from the b Now using the fact that wheel axes must intersect at a point when the mobile robot turns Thus we get tan φ1 = (c + d ) tan φ (c + d ) − b tan φ tan φ2 = (c + d ) tan φ (c + d ) + b tan φ and We introduce a vector of quasi velocities instead of generalized velocities because control runs in an abstract space Quasi-velocities are function of kinematic parameters Since the generalised velocities is always in the null space of A(q ) ,according to (1) the generalized velocities q can be expressed in terms of quasi-velocities v(t ) as follows q& = s (q ) v(t ) (2) where s ( q ) is a × full rank matrix, whose columns are 405 International Journal of Advanced Robotic Systems, Vol.5, No.4 (2008) in the null space of A(q ) and is given by ( using the MATHEMATICA package)  S11 S  21  S31  S s (q ) =  41 S  51  S61 S  71  S81 d   cos φ0 − c + d sin φ0 tan φ  x&0   d  y&     = sin φ0 + c + d cos φ0 tan φ φ&0   tan φ  &  φ   c+d   0  0  0 0  0 0   S11 = (c + d ) cos φ0 cot φ − d sin φ0 S 21 = d cos φ0 + ( c + d ) cot φ sin φ0 −2b ( c + d ) + b cot 2φ − cos ec 2φ (b + 2(c + d ) ) r (c + d − b tan φ ) + S 41 = (c + d ) (b − (c + d ) cot φ ) (c + d ) cos ecφ sec φ + b(2(c + d ) + b tan φ ) r (c + d + b tan φ ) + (c + d ) (b + (c + d )cot φ )2 −b + (c + d )cot φ r b + (c + d )cot φ S61 = r S71 = V2 = S81 = x&0 = vx cos φ0 − v y sin φ0 y& = vx sin φ0 + v y cos φ0 where v x and v y are the velocities of the centre of gravity of the mobile platform along the Thus we get x and y-axes respectively d  cosφ0 − sinφ0 tanφ  c+d  d  sinφ0 + cosφ0 tanφ  c+d  2  −2b(c + d ) + b cot 2φ − cos ec2φ (b + 2(c + d ) ) tanφ ⋅  x&0   c+d (c + d )2  y&   r (c + d − b tanφ ) +  0  (b − (c + d )cot φ )2 θ&1    &   (c + d )2 cos ecφ secφ + b(2(c + d ) + b tanφ ) tanφ ⋅ θ2  =  c+d (c + d )2 θ&   r (c + d + b tanφ ) +  3  φ ( b + ( c + d )cot ) θ&4   φ&   −b tanφ + (c + d )  0  r(c + d )  φ&    b tanφ + (c + d )  r(c + d )   tanφ  c +d   − s in φ s in φ u1 + (c + d ) c o s φ c o s φ u (c + d ) co s φ0 Then from above transformation we get the following two input four state chained form x& = u x& = u (4) x& = x u x& = x u where x = ( x1 , x2 , x3 , x4 ) is the state and u1 and u2 are the two control inputs  0  0   0     0 vx    φ&      0   0   0  1 Since the control objective for the robot is to ensure that q (t ) tracks a reference position and orientation denoted by qd (t ) = [ xd (t ), yd (t ),φd (t )] so we consider only 406 2.6 Chained form In order to convert the kinematic model of the mobile robot in chained form following change of co-ordinates is used x1 = x − d c o s φ ta n φ x2 = (c + d ) co s3 φ x3 = ta n φ x = y − d s in φ Together with two input transformations u V1 = cos φ0 S51 = Define (3) Where v1 = vx and v2 = φ& Where S31 =  0    v1   v2  0   1 Design of trajectory tracking controller Denote the tracking error as xe = x − xd The error differential equation are x&e1 = u1 − ud x&e = u2 − ud x&e3 = xe 2ud + x2 (u1 − ud ) x&e = xe3ud + x3 (u1 − ud ) (5) The goal is to find a , Lipschitz continuous time-varying state-feedback controller u% i.e u  u =   = u% ( xe , ud , ud ) u2  such that the tracking error asymptotically, i.e xe converges to zero lim x − xd = under t →∞ appropriate conditions on the reference control functions ud and ud and initial tracking errors xe (0) ,with a good choice of λ Umesh kumar & Nagarajan sukavanam: Backstepping based trajectory tracking control of a four wheeled mobile robot We first introduce a change of coordinates and rearrange system (5) into a triangular –like form so that the integrator backstepping can be applied Denote x% d = ( xd , xd ) and let η (.; x%d ) : R → R be the mapping defined by ζ i = xe(4 − i +1) − ( xe ( n − i ) + xd ( n − i ) ) xe1 , ≤ i ≤ ζ = ζ − α1 (ζ ) Define Differentiating the function 2 V2 = V1 + ζ = ζ + ζ 2 2 along the solution of (6) yields V&2 = ud 1ζ ζ − x2 ζ (u1 − ud )ζ − u2 ζ 2ζ ζ = xe ζ = xe1 As ζ = ζ − α1 (ζ ) In the new coordinates ζ = (ζ , ζ , ζ , ζ ) system (5) is ζ = ζ2− transformed into = ζ = ud 1ζ − u2ζ ζ&1 = ud1ζ − x2 (u1 − ud1)ζ = ud 1ζ − u2ζ + ud 1ζ − ud 1ζ ζ&2 = ud1ζ − u2ζ ζ&3 = u2 − ud = ud (ζ + ζ ) − u2ζ − ud 1ζ (6) ζ&4 = u1 − ud1 The basic design idea for backstepping based control law is to take for every lower dimension subsystem, some state variables as virtual control inputs and at the same time ,recursively select an appropriate Lyapunov function candidate Thus each step results in a new virtual control -er In the end of the overall procedure ,the true control law results which achieves the original design objective Consider the ζ − subsystem of (6) = ud (ζ − α ) − u2ζ − ud 1ζ Where α (ζ , ζ ) = −ζ Again define ζ = ζ − α (ζ , ζ ) ζ = ζ3− ( ζ&1 = ud 1ζ − x2 (u1 − ud )ζ the variable ud and ζ as time varying functions Denote ζ = ζ Differentiating the function V1 = ζ along the solution ∂α ∂α ζ + 2ζ ) ∂ζ 1 ∂ζ 2 = ζ − ( −ζ ) = ζ + ζ (7) We consider the variable ζ as virtual control input and = u2 − ud + ud 1ζ − x2 (u1 − ud )ζ Consider the positive definite and proper function 2 2 V3 = V2 + ζ = ζ + ζ + ζ 2 2 Differentiating the function V3 along the solution of (6) yields of (6) yields V&1 = ud 1ζ 1ζ − x2 ζ (u1 − ud )ζ V&3 = ζ (ud 1ζ + u2 − ud + ud 1ζ ) Selecting ζ = − k1ζ k1 ≥ whenever ζ = From above equation we observe that function α1 (ζ ) = is a stabilizing function i.e the desired value of virtual control ζ for which V1 is negative semi definite for subsystem (7) This desired value is called stabilizing function and ζ is called virtual control As in this case V& = −2k u V − ( x2 ζ + x2 ζ )(u1 − ud )ζ − u2 ζ 2ζ (8) Since the variable ζ is a virtual control input V&1 = −k1ud21ζ ∂α1 ζ ∂ζ 1 d1 Above implies that In order to make V3 negative definite we choose the following control input ud 1ζ + u2 − ud + ud 1ζ = −c3 ζ (9) where c3 > Thus we have V&3 = −c3 ζ − ( x2 ζ + x2 ζ )(u1 − ud )ζ − u2 ζ 2ζ Finally consider the positive definite and proper function which serves as a candidate Lyapunov function for the whole system (6) So the origin ζ = is asymptotically stable Hence the 2 λ ζ4 2 2 Where λ > is a design parameter Differentiating the function V4 along the solution of function α1 (ζ ) = is a stabilizing function (6) yields V1 → as t → ∞ ξ1 → as t → ∞ i.e V1lim = V4 = V3 + λ ζ 42 = ζ + ζ + ζ + 407 International Journal of Advanced Robotic Systems, Vol.5, No.4 (2008) V&4 = −c3 ζ − ( x2 ζ + x2 ζ )(u1 − ud )ζ − u2 ζ 2ζ + λζ (u1 − ud ) 2.5 V&4 = −c3 ζ + [{λ − ( x2 ζ + x2 ζ )}(u1 − ud ) − u2 ζ ]ζ In order to make V4 negative definite we choose the following control input 1.5 {λ − ( x2 ζ + x2 ζ )}(u1 − ud ) − u2 ζ = −c4ζ (10) From (9) and (10) we get the following control law 0.5 −c4ζ + u2ζ λ − (2 x2ζ + x2ζ ) u2 = ud − c3 (ζ + ζ ) − ud 1ζ u1 = ud + 10 20 30 40 50 secs Fig.4: norm of tracking error xe (t ) versus time Simulation results: To examine the effectiveness of the proposed trajectory tracking control methodology, the simulation for a four wheeled mobile robot were performed in MATHEMATICA The system parameters of the four wheel mobile robot were selected as c=1.3m, d=1.4 m, 2b=1.5m 4.1.Tracking a straight line Tracking a straight line is a simple case for all robots.Here for simulation we consider that the straight line.We choose the following as design parameters λ =5,c3 = c4 = for tracking a straight line It is further 0.2 0.1 -0.1 -0.2 10 15 20 assumed that initially xe (0) = (2,0.8,0.8,0.8) The straight line reference trajectory to be tracked is given by xd (t ) = t , yd (t ) = The desired trajectory and the norm of the tracking error is shown in fig and fig.4 respectively Fig.5:Desired sinusoidal trajectory we consider the following desired sinusoidal trajectory xd (t ) = t , yd (t ) = a sin ω t as shown in fig 10 desired steering angle 4.2.Tracking the curve xd (t ) = t , yd (t ) = a sin ωt 0.1 0.05 -0.05 -0.1 10 secs 15 20 0 secs Fig.3:desired straight line trajectory 10 Fig.6 Desired steering angle versus time Fig.7 demonstrates the evolution of the norm of the tracking error xe (t ) based on the following choice of design parameters and initial condition: λ =3,c3 = c4 = 5, xe (0) = (2,0.5,0.5,1.5) 408 Umesh kumar & Nagarajan sukavanam: Backstepping based trajectory tracking control of a four wheeled mobile robot with a=0.2 and ω =0.5 2.5 1.5 0.5 10 20 30 secs 40 50 Fig.7: Norm of tracking error xe (t ) versus time Conclusion In this paper the nonholonomic constraints and the kinematics model of the four wheel (front steering and rear driving) mobile robot under pure rolling and no side slipping condition is derived Using the change of coordinates the system is transformed into chained form and then a backstepping based tracking controller is derived Simulation results are presented with two examples to illustrate the approach Acknowledgement: The author Umesh Kumar gratefully acknowledges the financial support of University grant commission (UGC),Government of India through senior research fellowship with grant No.6405-11-61 The author Nagarajan Sukavanam acknowledges financial support by the Department 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