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Hybrid Control Design for a Wheeled Mobile Robot

Thomas Bak1 , Jan Bendtsen1 , and Anders P. Ravn2

1

Department of Control Engineering, Aalborg University, Fredrik Bajers Vej 7C, DK-9220 Aalborg, Denmark, {tb,dimon}@control.auc.dk 2 Department of Computer Science, Aalborg University, Fredrik Bajers Vej 7E, DK-9220 Aalborg, Denmark, apr@cs.auc.dk

Abstract. We present a hybrid systems solution to the problem of trajectory tracking for a four-wheel steered four-wheel driven mobile robot. The robot is modelled as a non-holonomic dynamic system subject to pure rolling, no-slip constraints. Under normal driving conditions, a nonlinear trajectory tracking feedback control law based on dynamic feedback linearization is suf?cient to stabilize the system and ensure asymptotically stable tracking. Transitions to other modes are derived systematically from this model, whenever the con?guration space of the controlled system has some fundamental singular points. The stability of the hybrid control scheme is ?nally analyzed using Lyapunov-like arguments. Keywords: hybrid systems, robot, trajectory tracking, stability analysis

1 Introduction

Wheeled mobile robots is an active research area with promising new application domains. Mobile robots are mechanical systems characterized by challenging (nonintegrable) constraints on the velocities which have lead to numerous interesting path tracking control solutions, [12], [11], [4]. Recently [3] and [1] have addressed the problem from a Hybrid systems perspective. We consider a problem of similar complexity and develop a systematic approach to derivation of a hybrid automaton and to stability analysis. Our work is motivated by a project currently in progress, where an autonomous four-wheel driven, four-wheel steered robot (Figure 1) is being developed. The project needs a robot that is able to survey an agricultural ?eld autonomously. The vehicle has to navigate to certain waypoints where measurements are taken of the crop and weed density. This information is processed and combined to a digital map of the ?eld, which eventually will allow the farm manager to manage weed infestations in a spatially precise manner. The robot is equipped with GPS, gyros, magnetometer and odometers, which will not only help in the exact determination of the location where each image is taken, but also provide measurements for an estimate of the robot’s position and orientation for a tracking algorithm. Actuation ia achieved using independent steering and drive motors (8 brushless DC motors in total). The robot navigates from waypoint to waypoint following spline-type trajectories between the waypoints to minimize damage to the crop. From a control point of view, this is a tracking problem. To solve this problem we ?rst derived a dynamics model of

Fig. 1. Schematic model of the experimental platform. The robot is equipped with 8 independent steering and drive motors. Localization is based on fusion of GPS, gyro, magnetometer, and odometer date.

the vehicle subject to pure rolling, no-slip constraints, following the approach taken in [5] and [6]. Next we design a path tracking control law based on feedback linearization. Feedback linearization designs have the potential of reaching a low degree of conservativeness, since they rely on explicit cancelling of nonlinearities. However, such designs can also be quite sensitive to noise, modelling errors, actuator saturation, etc. As pointed out in [8], uncertainties can cause instability under normal driving conditions. This instability is caused by loss of invertibility of the mapping representing the nonlinearities in the model. Furthermore, there are certain wheel and vehicle velocity con?gurations that lead to similar losses of invertibility. Since these phenomena are, in fact, linked to the chosen control strategy rather than the mechanics of the robot itself, we propose in this paper to switch between control strategies such that the aforementioned stability issues can be avoided. We intend to motivate the rules for when and how to change between the individual control strategies directly from the mathematical-physical model. We will consider the conditions under which the description may break down during each step in the derivation of the model and control laws. These conditions will then de?ne transitions in a hybrid automaton that will be used as a control supervisor. However, introducing a hybrid control scheme in order to improve the operating range where the robot can operate in a stable manner comes at a cost: The arguments for stability become more complex. Not only must each individual control scheme be stable; they must also be stable under transitions. As one of the main contributions of this work, We will therefore apply the generalized Lyapunov stability theory as introduced by Branicky in [2]. Even when given suitable Lyapunov functions for each mode, reasoning about overall stability is still a major effort. We thus abstract the Lyapunov functions to a constant rate function, where the rate is equivalent to the convergence rate. Each mode or state

of the original automaton is then replaced by three consecutive states. The ?rst of these states models the initial transition cost and settling period where the function may increase, albeit for a bounded time, while the second and third state models the working mode with the local Lyapunov function. The third state is the transition safe state, where the Lyapunov function has decreased below its entry value. All three states are guarded by the original conditions for a mode change; but it is potentially unsafe to leave before the third state is entered. A straightforward analysis will show that the system can always be rendered unstable: Just vary the reference input such that transitions are always taken before the transition safe state. We thus add a second automaton constraining the change of the reference input (the trajectory) such that the resulting system remains stable. This automaton de?nes safe operating conditions, or put another way: Constraints to be satis?ed by the trajectory planner. The composed automaton is in a form, such that it can be analyzed using model checking tools.

2 Dynamic Model and Linearization

In the following we derive the model and the normal mode control scheme. During the derivation we note conditions for mode changes. We consider a four-wheel driven, four-wheel steered robot moving on a horizontal plane, constructed from a rigid frame with four identical wheels. Each wheel can turn freely around its horizontal and vertical axis. The contact points between each of the wheels and the ground must satisfy pure rolling and non-slip conditions.

ICR yF 6

M yv

Wheel 2 y

1

Wheel 1 γ1 θ

1xv

Wheel 4

Wheel 3

x

- xF

Fig. 2. De?nition of the ?eld coordinate system (xF , yF ), vehicle coordinate system (xv , yv ), vehicle orientation θ, distance 1 , and direction γ1 from the center of mass (x, y ) to wheel 1. Each wheel plane is perpendicular to the Instantaneous Center of Rotation (ICR).

Consider a reference (‘?eld’) coordinate system (xF , yF ) in the plane of motion as illustrated in Figure 2. The robot position is then completely described by the coordinates (x, y ) of a reference point within the robot frame, which without loss of generality

can be chosen as the center of mass, and the orientation θ relative to the ?eld coordinate system of a (‘vehicle’) coordinate system (xv , yv ) ?xed to the robot frame. These coordinates are collected in the posture vector ξ = [x y θ]T ∈ R2 × S1 . The position of the i’th wheel (1 ≤ i ≤ 4) in the vehicle coordinate system is characterized by the angle γi and the distance i . As the wheels are not allowed to slip, the planes of each of the wheels must at all times be tangential to concentric circles with the center in the Instantaneous Center of Rotation (ICR). The orientation of the plane of the i’th wheel relative to xv is denoted βi . The vector β = [β1 β2 β3 β4 ]T ∈ S4 de?ne the wheel orientations. From an operational point of view a relevant speci?cation of the ICR is to give the orientation of two of the four wheels. We therefore partition β into βc ∈ S2 containing the coordinates used to control the ICR location and βo ∈ S2 containing the two remaining coordinates that may be derived from the ?rst. Cross Driving (Singular Wheel Con?guration) An important ambiguity (or singular wheel con?guration), is present in the approach taken above. For β1 = ±π/2 and β2 = ±π/2 the con?guration of wheels 3 and 4 is not de?ned. The situation corresponds to the ICR being located on the line through wheel 1 and 2. The wheel con?guration βc = [β3 β4 ]T result in similar problems and both con?gurations fail during cross driving as ˙ = 0). To ensure safe solutions to the all wheels are at ±π/2 i.e. ICR at in?nity (and θ trajectory tracking problem we must ensure that the singular con?gurations are avoided at all times. Based on this discussion we identify three discrete control modes, q1 , q2 and q3 : q1 : Trajectory tracking with βc = [β1 β2 ]T . This mode is conditioned on |β1 | < (π/2 ? ˙ | ≥ o. b) ∧ |β2 | < (π/2 ? b) ∧ |θ q2 : Trajectory tracking with βc = [β3 β4 ]T . This mode is conditioned on |β3 | < (π/2 ? ˙ | ≥ o. b) ∧ |β4 | < (π/2 ? b) ∧ |θ ˙ | < o. q3 : Cross Driving with β1 = β2 = β3 = β4 . This mode is conditioned on |θ where b and o are small positive numbers. The two ?rst modes cover the situations where the ICR is governed by wheels 1 and 2 and by wheel 3 and 4, respectively, while the latter covers the situation where the ICR is at in?nity. For brevity of the exposition, we will consider βc = [β1 β2 ]T in the following; the case with βc = [β3 β4 ]T is analogous. As presented here, the problem is speci?c to the proposed representation of the ICR, but in general, no set of two variables is able to describe all wheel con?gurations without singularities [13]. That is, the problem of such singular con?gurations is not due to the representation used, but is a general problem for this type of robotic systems. 2.1 Vehicle Model Following the argumentation in Appendix A, robot posture can be manipulated via one velocity input η (t) ∈ R in the instantaneous direction of Σ (βc ) ∈ R3 which is constructed to meet the pure-roll constraint. Similarly, it is possible to manipulate the ori˙1 β ˙ 2 ]T ∈ R2 . The entations of the wheels via an orientation velocity input ζ (t) = [β

no-slip condition on the wheels that constrain η (t) is handled (see Appendix A) by applying Lagrange formalism and computed torque techniques. The result is the following extended dynamical model: ? ? ? ? ? ? ˙ 0 RT (θ)Σ (βc ) 0 00 ξ ν 0 0? χ + ?1 0? ˙ ? = ?0 χ ˙ =?η ζ ˙c 0 0 0 0I β

(1)

where ν is a new exogenous input that is related to the torque applied to the drive motors. In equation (1) it is assumed that the β dynamics can be controlled via local ˙ as an exogenous input to the model. servo loops, such that we can manipulate β 2.2 Normal Trajectory Tracking Control Provided that we avoid the singular wheel con?gurations the standard approach from here on is to transform the states into normal form via an appropriate diffeomorphism followed by feedback linearization of the nonlinearities and a standard linear control design. We choose the new states x1 = T (χ) = which yields the following dynamics: x ˙ 1 = A1 x1 + B1 δ (χ) ν ? α(χ) , ζ A1 = 0I , 00 B1 = 0 . I (3) ξref ? ξ ˙ref ? ξ ˙ , ξ (2)

Using the results from Appendix A, δ (χ) and α(χ) may be found to δ (χ) = RT (θ) Σ (βc ) N (βc )η and ? ? ? 1 sin β2 cos(β1 ? γ1 ) + 2 sin β1 sin(β2 ? γ2 ) α(χ) = sin(β1 ? β2 )η 2 ? 1 cos β2 cos(β1 ? γ1 ) ? 2 cos β1 cos(β2 ? γ2 ) ? 0 (5) (4)

where N (βc ) = [N1 N2 ] is speci?ed in equations (19) and (20). If we then apply the control law ν = δ (χ)?1 (α(χ) ? K1 x1 ) (6) ζ we obtain the closed-loop dynamics x ˙ 1 = (A1 ? B1 K1 )x1 , which tends to 0 as t → ∞ if K1 is chosen such that A1 ? B1 K1 has eigenvalues with negative real parts. Similar dynamics can be obtained for the situation with βc = [β3 β4 ]T , resulting in closed-loop dynamics x ˙ 2 = (A2 ? B2 K2 )x2 .

2.3 Parallel Wheels Control Due to the conditions imposed on the trajectory tracking controllers, an additional control scheme must be derived that is able to control the vehicle when all four wheels are ˙ to 0. Fortunately, the dynamics of the robot becomes particuparallel, which forces θ larly simple in this case. ˙ = 0 the dynamics are immediately linear; hence, choosing the states With θ x3 = T χ = ξref ? ξ ˙ref ? ξ ˙ , ξ (7)

where T is an appropriate invertible matrix, then yields the dynamics x ˙ 3 = A3 x3 + B3 ν , 0 A3 = 0 I , A31 A32 B3 = 0 I (8)

which can be controlled by applying the feedback ν = ?K3 x3 . Rest Con?gurations During the feedback linearization design we detect another interesting condition due to the inversion of δ (χ). If δ (χ) looses rank, the control strategy breaks down and the control input grows to in?nity. If we avoid the rest con?guration, η = 0, then Σ (βc ) speci?es the current direction of movement and the column vectors N1 and N2 are perpendicular to this direction and to each other. To avoid an ill-conditioned δ (χ) we must impose a new condition, |η | ≥ n, where n is a small positive number, on our trajectory tracking modes, q1 and q2 . To complete the construction, we add additional modes to handle the rest con?guration. First assume that the robot is started with β1 = β2 = β3 = β4 . We may then utilize the controller de?ned for the q3 mode, choosing ξref as an appropriate point on the straight line originating from the center of mass in the direction de?ned by β along with ˙ref = ηref = 2n ξ ζref 0 This mode would allow the robot to start from rest and adds a mode (q0 ). Finally we add a mode q4 to handle the stop situation. Again we assume that the wheels have been oriented by the control laws in mode q1 or q2 such that the waypoint lies on the straight line from the center of mass in the direction de?ned by β . We may then apply the same state transformation as in (7) along with the same state feedback, and choosing ξref ˙ref = 0. As a result |η | ≥ n is added to the mode as the target waypoint along with ξ conditions for q1 , q2 , and two new modes q0 and q4 are added to handle the start from rest and the stop situations respectively.

3 Hybrid Automaton Supervisor

The trajectory tracking problem for this particular robot may be solved by applying the different control laws, as outlined above for different modes. The conditions for exiting

the modes have been de?ned as well. For each of these modes, we de?ned special control schemes, and conditions. Given that there are two modes where the robot is at rest, and three modes where the robot is driving, it is straightforward to introduce two super-modes, Rest and Driving. This gives rise to the hierarchical hybrid automaton implemented using State?ow as shown in Figure 3.

Driving Rest /mode=Q4 [abs_beta[1]<BLIM & abs_beta[2]<BLIM & abs_bdot >= o] /mode=Q1 q1_left_wheels [abs_eta<=ELIM] q4_stop [abs_beta[3]<BLIM & abs_beta[4]<BLIM & abs_bdot >= o] /mode=Q2 [abs_bdot<o]/mode=Q3 [new_wp==1]/mode=Q0 [abs_beta[1]<BLIM & abs_beta[2]<BLIM & abs_bdot >= o] /mode=Q1 q2_right_wheels

q0_start [abs_eta>ELIM] q3_parallel_wheels [abs_bdot<o]/mode=Q3

/mode=Q3

[abs_beta[3]<BLIM & abs_beta[4]<BLIM & abs_bdot >= o] /mode=Q2

Fig. 3. State?ow representation of Automaton.

The hybrid automaton [10] consists of ?ve discrete states, Q = {q0 , q1 , q2 , q3 , q4 } as de?ned during the model and controller derivation. The continuous state x de?ned by equation (2) or (7) belongs to the state space X ? R2 × S1 × R3 . The corresponding hybrid state space is H = Q × X . The vector ?elds are de?ned by ? ? (A3 ? B3 K3 )x0 ? ? ? ? ? ?(A1 ? B1 K1 )x1 f (q, x) = (A2 ? B2 K2 )x2 ? ? ?(A3 ? B3 K3 )x3 ? ? ? ?(A ? B K )x 3 3 3 4 if q if q if q if q if q = q0 , = q1 , = q2 , = q3 , = q4 .

(9)

Conditions and guards are given in Figure 3 based on the derivations in Section 2. The system including the supervisor was simulated in Simulink and the tracking of an example trajectory is shown in Figure 4. The system is clearly able to start from an rest con?guration, track the trajectory and stop at a rest con?guration. In this example the controller starts in the mode q4 , switches to a new waypoint and trajectory information becomes available. As η grows the mode is changed to q3 and eventually q1 where it remains until the vehicle returns to q4 and stops. Mode changes, tracking errors and wheel positions are given in Figure 5.

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Fig. 4. Tracking a reference trajectory. The vehicle is initially at rest offset from the trajectory by 1 meter in the y-direction.

1 βc [rad/s] 0

?1 pos. error [m] 1 0.5 0 0.5 0 ?0.5 5 0 ?5 q4 q3 q2 q1 q0 η

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angle error [rad]

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Fig. 5. Wheel orientation (β1 , β2 ), tracking errors and modes for the case given in Figure 4.

4 Stability Analysis

In order to ensure that the hybrid control law speci?ed in the previous Section does not cause the robot to become unstable, we will carry out a stability analysis of the closed loop system. Here, we will use the notion of stability of switched systems introduced in [2], which is summarized in Appendix B. In case of the autonomous robot, we have the following Lyapunov functions for the individual control modes. q0 : q1 : q2 : q3 : q4 : Starting from rest with β1 = β2 = β3 = β4 : V0 (x0 ) = (x0 )T P3 x0 . Trajectory tracking with βc = [β1 β2 ]T : V1 (x1 ) = (x1 )T P1 x1 Trajectory tracking with βc = [β3 β4 ]T : V2 (x2 ) = (x2 )T P2 x2 Cross driving with β1 = β2 = β3 = β4 : V3 (x3 ) = xT 3 P3 x3 Stopping with β1 = β2 = β3 = β4 : V4 (x4 ) = xT P x 4 3 4.

In each of the cases listed above, Pj = PjT > 0 is the positive de?nite solution to the Lyapunov equation Pj (Aj ? Bj Kj ) + (Aj ? Bj Kj )T Pj = ?I . Note that for modes q0 and q4 , the same state feedback K3 and solution matrix P3 as in mode q3 are used. Elementary calculations now yield ˙ j = (x V ˙ j )T Pj xj + xT ˙ j = ?xT j Pj x j xj . With this in place, we can now attempt to evaluate the Lyapunov functions for each mode using a hybrid automaton. We ?rst observe, that since we focus on stability only, we can in each mode abstract from the concrete evolution of the state and replace it by the evolution of the Lyapunov function. As mentioned in Section 1, we furthermore divide each discrete state qj , j = 0, . . . , 4 in the automaton in Figure 3 into three consecutive states that evaluate a constant rate variable which dominates the Lyapunov function. These states are: An entry state, which represents the gain in the Lyapunov function Vj (xj ) at the instant the hybrid control law switches to mode j ; an operation state, which represents the period where the feedback control ν = ?Kj xj is active, and where Vj (xj ) is decreasing toward 0; and a ?nal state, where Vj (xj ) is small enough to indicate that the system has become stable (i.e., the tracking error is suf?ciently small to ignore). The basic idea is depicted in Figure 4. When the control enters mode j , the Lyapunov function will have gained an amount ?Vj since the last time it was active. This is modelled abstractly as a constant rate function Vj (T ) = αp T + Vj (0), 0 ≤ T ≤ Tpenalty , αp ≥ 0, where the time Tpenalty is determined such that ?Vj = αp Tpenalty . Subsequently, the ˙ j is negative de?nite. Consequently, Vj (T ) operation state becomes active, in which V is bounded from above by the function V (T ) = ?αo T + ?Vj + Vj (0), Tpenalty ≤ T ≤ Tstable , αo ≥ 0, i.e., another constant rate automaton. Finally, when Tstable is reached, the system is considered stable and stays in this state with the trivial constant rate function Vstable (T ) = Vstable (0) until another mode change is enforced, e.g., because the target waypoint is reached. In State?ow the hierarchy is achieved using subcharted states, which allows us to divide the states in Figure 3 into three new substates. It is clear that only the transition from the ?nal state is guaranteed safe. To avoid

6

Vj (0) + ?Vj

Vj (0) T = 0 T = Tpenalty T = Tstable

- ’Time’ T

- q0

- q1

- q2

Fig. 6. Abstract three-state automaton simulating the Lyapunov function of mode j . The entry, operation and ?nal states are indicated below the ?gure.

unsafe transitions due to the input we propose to add a second automaton constraining the change of the reference input (the trajectory). This automaton has three states, startup, constant_speed and stop which allows ut to specify the basic operation of the path planning. The trajectory planner transitions conditions are guarded by the transitions in the automation describing the Lyapunov function. If all mode transitions from the two unsafe states (entry, operation) are redirected to an error mode, and the parallel composition with the path planner has error as an unreachable state – the system is safe.

5 Conclusion

We have developed a hybrid control scheme for a path-tracking four-wheel driven, fourwheel steered autonomous robot, and shown how it is analyzed for stability. The basis for controller development is standard non-slipping and pure rolling conditions, which are used to establish a kinematic-dynamical model. This is used to ?nd a partial linearization of the dynamics using computed torques and local servo loops around the steering motors. Then, a normal mode path tracking controller is designed according to the feedback linearization method. Other modes are introduced systematically, where the model has singularities. For each such case a transition condition and a new control mode is introduced. Specialized controllers are developed for such modes. With the control automaton completed, we found for each mode, Lyaponov-like functions, which combine to prove stability. In order to simplify the analysis, we bound the Lyapunov functions by constant rate functions. This allows us to show stability by analyzing a version of the control automaton, where each mode contains a simple three state automaton that evaluates the constant rate functions. Discussion and Further Work In the systematic approach to deriving modes, we list conditions when the normal mode model fails. Some of these, e.g. Cross Driving, are rather obvious when developing the model; but others, e.g. the Rest Con?guration, are less clear, because they are not outright singularities, but more conditions that make the model ill conditioned. Such problems are usually detected during simulation. Thus

a practical rendering of the systematic approach is to use a tool like State?ow and build the normal mode model. When the simulation has problems, one investigates the conditions and de?nes corresponding transitions. An approach that we believe is widely applicable to design of supervisory or mode switched control systems. Such a divide and conquer approach is evidently only safe to the extent that it is followed by a rigorous stability analysis.The approach which we develop is very systematic. It ends up with a constant rate hybrid automaton which should allow model checking of its properties. In particular, whether it avoids unsafe transitions when composed with an automaton modelling the reference input. A systematic analysis of this combination is, however, future work. Another point that must be investigated is, how the wheel reference output is made bumpless during mode transitions.

References

1. C. Alta?ni, A. Speranzon, K.H. Johansson. Hybrid Control of a Truck and Trailer Vehicle, In C. Tomlin, and M. R. Greenstreet, editors, Hybrid Systems: Computation and Control LNCS 2289, p. 21ff, Springer-Verlag, 2002. 2. M. S. Branicky. Analyzing and Synthesizing Hybrid Control Systems In G. Rozenberg, and F. Vaandrager, editors, Lectures on Embedded Systems, LNCS 1494, pp. 74-113, SpringerVerlag, 1998. 3. A. Balluchi, P. Souères, and A. Bicchi. Hybrid Feedback Control for Path Tracking by a Bounded-Curvature Vehicle In M.D. Di Benedetto, and A.L. Sangiovanni-Vincentelli, editors, Hybrid Systems: Computation and Control, LNCS 2034, pp. 133–146, Springer-Verlag, 2001. 4. G. Bastin, G. Campion. Feedback Control of Nonholonomic Mechanical Systems, Advances in Robot Control, 1991 5. B. D’Andrea-Novel, G. Campion, G. Bastin. “Modeling and Control of Non Holonomic Wheeled Mobile Robots, in Proc. of the 1991 IEEE International Conference on Robotics and Automation, 1130–1135, 1991 6. G. Campion, G. Bastin, B. D’Andrea-Novel. “Structural Properties and Classi?cation of Kinematic and Dynamic Models of Wheeled Mobile Robots, IEEE Transactions on Robotics and Automation Vol. 12, 1:47-62, 1996 7. L. Caracciolo, A. de Luca, S. Iannitti. “Trajectory Tracking of a Four-Wheel Differentially Driven Mobile Robot, in Proc. of the 1999 IEEE International Conference on Robotics and Automation, 2632–2838, 1999 8. J.D. Bendtsen, P. Andersen, T.S. Pedersen. “Robust Feedback Linearization-based Control Design for a Wheeled Mobile Robot, in Proc. of the 6th International Symposium on Advanced Vehicle Control, 2002 9. H. Goldstein. Classical Mechanics, Addison-Wesley, 2nd edition, 1980 10. T. A. Henzinger. The Theory of Hybrid Automata? In Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science (LICS 1996), pp. 278-292, 1996. 11. C. Samson. Feedback Stabilization of a Nonholonomic Car-like Mobile Robot, In Proceedings of IEEE Conference on Decision and Control, 1991. 12. G. Walsh, D. Tilbury, S. Sastry, R. Murray, J.P. Laumond. Stabilization of Trajectories for Systems with Nonholonomic Constraints IEEE Trans. Automatic Control, 39: (1) 216-222, 1994 13. B. Thuilot, B. D’Andrea-Novel, A. Micaelli. “Modeling and Feedback Control of Mobile Robots Equipped with Several Steering Wheels, IEEE Transactions on Robotics and Automation Vol. 12, 2:375-391, 1996

A Vehicle Dynamics

Denote the rotation coordinates describing the rotation of the wheels around their horizontal axes by φ = [φ1 φ2 φ3 φ4 ]T ∈ S4 and the radii of the wheels by r = [r1 r2 r3 r4 ] ∈ R4 . The motion of the four-wheel driven, four-wheel steered robot is then completely described by the following 11 generalized coordinates: q = x y θ β T φT

T

= ξ T β T φT

T

(10)

and we can write the pure rolling, no slip constraints on the compact matrix form A(q )q ˙= in which J1 (β )R(θ) 0 J2 q ˙=0 C1 (β )R(θ) 0 0 (11)

? ? cos β1 sin β1 1 sin(β1 ? γ1 ) ?cos β2 sin β2 2 sin(β2 ? γ2 )? ? J1 (β ) = ? ?cos β3 sin β3 3 sin(β3 ? γ3 )? , J2 = rI4×4 , cos β4 sin β4 4 sin(β4 ? γ4 ) ? ? ? ? ? sin β1 cos β1 1 cos(β1 ? γ1 ) cos θ sin θ 0 ?? sin β2 cos β2 2 cos(β2 ? γ2 )? ? ? ? C1 ( β ) = ? ?? sin β3 cos β3 3 cos(β3 ? γ3 )? , and R(θ) = ? sin θ cos θ 0 . 0 0 1 ? sin β4 cos β4 4 cos(β4 ? γ4 )

˙ is constrained to belong Following the argumentation in [6], the posture velocity ξ to a one-dimensional distribution here parametrized by the orientation angles of two wheels, say, β1 and β2 . Thus, ˙ ∈ span{col{R(θ)T Σ (βc )}} ξ where Σ (βc ) ∈ R3 is perpendicular to the space spanned by the columns of C1 , i.e., C1 (β )Σ (βc ) ≡ 0 ?β . Σ can be found by combining the expression for C1 (β ) with equations for the orientation of wheels 3 and 4 to ? ? 1 cos β2 cos(β1 ? γ1 ) ? 2 cos β1 cos(β2 ? γ2 ) Σ = ? 1 sin β2 cos(β1 ? γ1 ) ? 2 sin β1 cos(β2 ? γ2 ) ? . sin(β1 ? β2 ) The discussion above implies that the robot posture can be manipulated via one ˙(t) = velocity input η (t) ∈ R in the instantaneous direction of Σ (βc ), that is, R(θ)ξ Σ (βc )η (t) ?t. Similarly, it is possible to manipulate the orientations of the wheels via ˙1 β ˙ 2 ]T ∈ R2 . an orientation velocity input ζ (t) = [β The constrained dynamics of η are handled by applying Lagrange formalism and computed torque techniques as suggested in [5] and [6]. The Lagrange equations for non-holonomic systems are written on the form [9] d dt ?T ?q ˙k ? ?T = ck (q )T λ + Qk ?qk

in which T is the total kinetic energy of the system and qk is the k ’th generalized coordinate. On the left-hand side, ck (q ) is the k ’th column in the kinematic constraint matrix A(q ) de?ned in (11), λ is a vector of so-called Lagrange undetermined coef?cients, and Qk is a generalized force (or torque) acting on the k ’th generalized coordinate. The kinetic energy of the robot is calculated as ? ? R(θ)T M R(θ) R(θ)T V 0 1 T? V T R(θ) Jβ 0 ?q ˙ (12) T = q ˙ 2 0 0 Jφ with appropriate choices of M , Jβ and Jφ . In the case of the wheeled mobile robot we can derive the following expressions: ? ? 4 m f + 4m w 0 ?mw i=1 i sin γi 4 (13) M =? 0 mf + 4mw mw i=1 i cos γi ? . 4 4 4 2 ?mw i=1 i sin γi mw i=1 i cos γi If + mw i=1 γi Here, If is the moment of inertia of the frame around the center of mass, and mf and mw are the masses of the robot frame and each wheel, respectively. We note that since the wheels are placed symmetrically around the xv and yv axes, the off-diagonal terms should vanish. However, this may not be possible to achieve completely in practice, due to uneven distribution of equipment within the robot. Turning to the wheels, we denote the moment of inertia of each wheel by Iw and ?nd 1 Jβ = Iw I4×4 and Jφ = Iw I4×4 (14) 2 and ? ? 0 0 0 0 V = ? 0 0 0 0 ?. (15) Iw Iw Iw Iw The Lagrange undetermined coef?cients are then eliminated in order to arrive at the following dynamics: h1 (β )η ˙ + Φ1 (β )ζη = Σ T Eτφ (16)

T ?1 J2 ∈ R3×4 and τφ ∈ R4 is a vector of torques applied to drive the in which E = J1 wheels. The quadratic function h1 (β ) is given by

h1 (β ) = Σ T (M + EJφ E T )Σ > 0 and Φ1 (β ) ∈ R is given by Φ1 (β ) = Σ T (M + EJφ E T )N (βc ) N (βc ) = [N1 N2 ], where ? ? ? 1 cos β2 sin(β1 ? γ1 ) + 2 sin β1 cos(β2 ? γ2 ) N1 = ?? 1 sin β2 sin(β1 ? γ1 ) ? 2 cos β1 cos(β2 ? γ2 )? cos(β1 ? β2 ) ? ? ? 1 sin β2 cos(β1 ? γ1 ) + 2 cos β1 sin(β2 ? γ2 ) N2 = ? 1 cos β2 cos(β1 ? γ1 ) + 2 sin β1 sin(β2 ? γ2 ) ? ? cos(β1 ? β2 )

(17)

(18)

(19)

(20)

Equation (16) can be linearized by using a computed torque approach and choosing τφ appropriately. The torques are simply distributed evenly to each wheel; we observe that ? ? τ1 ? τ2 ? ? Σ T Eτφ = [a1 a2 a3 a4 ] ? ?τ3 ? = L τ4 where L is the left-hand side of (16). Then we set τφ = Hτ0 , H ∈ R4 and choose Hi = Lsign(ai )/σ , where σ is the sum of the four entries in the vector Σ T E . This distribution policy ensures that the largest torque applied to the individual wheels is as small as possible. Hence, by applying the torque τ0 = we obtain η ˙=ν where ν is a new exogenous input. The result of the extension and partial linearization is the dynamical model given in Equation 1. 1 (h1 (β )ν + Φ1 (β )ζη ) , Σ T EH (21)

B Stability of Switched Systems

Consider a dynamic system whose behavior at any given time t ≥ t0 , where t0 is an appropriate initial time, is described by one out of several possible individual sets of continuous-time differential equations Σ0 , Σ1 , . . . , Σ? , and let x0 (t), x1 (t), . . . , x? (t) denote the corresponding state vectors for the individual systems: Σj : x ˙ j = fj (xj (t)), j = 0, 1, . . . , ?

The governing set of differential equations is switched at discrete instances ti , i = 0, 1, 2, . . . ordered such that ti < ti+1 ?i. That is, the system behavior is governed by Σj in the time interval ti < t ≤ ti+1 , then by Σk in the time interval ti+1 < t ≤ ti+2 , and so forth. Assume furthermore that for each Σj there exists a Lyapunov function, ˙ (xj ) ≤ 0 for i.e., a scalar function Vj (xj (t)) satisfying Vj (0) = 0, Vj (xj ) ≥ 0, and V xj = 0. It is noted that, by the last requirement, Vj is a non-increasing function of time in the interval where Σj is active. Hence, it can be deduced that the switched system governed by the sequence of sets of differential equations is stable if it can be shown that Vj (xj (tq )) ≥ Vj (xj (tr )) for all 0 ≤ j ≤ ? and tq , tr ∈ {ti }, where tq < tr are the last and current switching time where Σj became active, respectively.

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