**6.4 Application in vehicle control**

**6.4.1 Control of caster-supported carts with two driving wheels**

In the past decade motion control of WMRs consisting of two actively driven wheels and a caster
ob-tained considerable attention (e.g. [76, 83, 84]). These approaches used Lyapunov’s 2nd method and
normally were based on complicated mathematical details. Our aim is to show that the RFPT-based
adaptive controller design allows the selection of*kinematically prescribed point of the WMR*for
con-trolling its motion and the rotation of the cart around this point. This point may differ from the mass
center point of the cart that was a very popular choice for tracking due to the fact that the equations
of motion in this case appear as that of a simple LTI system. If the tracked point is different to the
mass center point in the equations of motion strongly coupled nonlinear terms appear that makes the

controller design task very complicated. It is also shown that if the properties of the driving motors as Electrodynamic subsystems are also taken into consideration the control task becomes a third order one. It is also shown that by the use of this adaptive technique the order of the control task can be re-duced from 3 to 2. In this way I made a step towards the elaboration of a more general order reduction technique in the case of nonlinear systems.

The main problem in the control of such systems [A. 17]:

1. Kinematic constraints 1: the cart must move on the surface of a plain ground, the*independent*
*variables to be controlled*are the(x, y)coordinates of a particular point of the cart on this plain,
and the rotational position around the horizontal axis*θ.*

2. Kinematic constraints 2: the above condition allows skidding/sliding/slipping of the cart on the
ground. These effects must be excluded, i.e. *the system to be controlled is a non-holonomic*
*device*in which*the rotational speeds of the wheels uniquely determine the speed of the motion*
*over the ground.*

3. For solving this task we*have only two control agents, the torques exerted by the driven wheels.*

4. The dynamic model of the cart has simple equations*only if the the tracked point and the mass*
*center point of the system are identical. This condition rarely can be met.*

5. Normally the dynamic parameters of the system are*only approximately known.*

6. On the basis of Classical Mechanics a 2nd order control can be formulated for the rotation of the
wheels. *If the system is driven by electric DC motors the necessary torque components cannot*
*be instantaneously set, only the time-derivatives of these torque values can be instantaneously*
*prescribed.*

7. Consequently

(a) *either a 3rd order controller can be designed for the rotation of the wheels, or*
(b) *some order reduction technique must be elaborated for a nonlinear system*

Figure 6.48: The kinematic structure of the two wheels model in which only the wheels must remain in contact with the ground (rotations and torque components are deﬁned according to the “right handed convention“) [A. 17]

The solution is to 1. Local Optimization without Riccati Equations.The general conditions allow
two rotary degree of freedom. Let apply a*rotation of the cart around axlex*ˆ1*by angleq**u*radian by the
rotational matrix*U*(q* _{u}*)generated by the generator

*GU*ˆ as [A. 17]

*U*(q* _{u}*)

*= If the axles of the wheels in the “basic position” were parallel with*

^{def}*x*ˆ

_{1}this operation can be realized because it keeps the wheels in contact with the plane of motion that is perpendicular to

*x*ˆ

_{3}[A. 17].

Following that apply a*rotation aroundx*ˆ_{3}*with angleq** _{v}*radians by the orthogonal matrix

*V*(q

*) gen-erated by the generator*

_{v}*G*ˆ

^{(V}

^{)}as [A. 17]

This operation is possible, too, since it moves the wheels on the surface of the horizontal plain.
There-fore a two-parameters subgroup of the 3D rotational group was so found that its elements describe
the possible motion of the cart if it remains in contact with the horizontal plain. (q*v*≡*θ) [A. 17].*

From that fact cart is a Non-Holonomic device further constraints originate.

Since the rotation*U*(q*u*)around*x*1does not concern the position of the wheels on the ground only
*V*(q* _{v}*)is interesting that moves the wheels on the ground. (In the case of a caster

*U*≡

*I*.) [A. 17].

The non-holonomic constraints originate form the next ﬁgure:

Figure 6.49: The nature of the kinematic constraints in the case of a non-holonomic device [A. 17]

*R*˙1=*r**w**q*˙* _{r}*+ ˙

*q*

_{l}2 sinq*v**,R*˙2=−r*w**q*˙* _{r}*+ ˙

*q*

*2 cos*

_{l}*q*

*v*

˙

*q**v*=−^{r}_{D}^{w}^{q}^{˙}^{r}^{−}_{2}^{q}^{˙}^{l}*.* (6.31)

Apply a generally not realizable PID-type tracking policy by the use of the quantities to be controlled
Then in each program cycle the*allowedq*¨^{Des}_{r}*andq*¨^{Des}* _{l}* values that yield the best approximation of
the desired 2nd derivatives if a quadratic goal function

Φ( ¨*q*^{Des}_{r}*,q*¨^{Des}* _{l}* )

*=*

^{def}Suggested solution for simultaneous compensation of the modeling imprecisions and nonlinear order reduction.

Use the analytical form and the numerical parameters of the best available model for the cart’s dynamics! In our case it is [A. 17]:

*Iθ*¨=_{2D}* ^{Ir}* ( ¨

*q*

*−*

_{r}*q*¨

*) =*

_{l}

^{D}*(T*

_{r}*−*

_{r}*T*

*),*

_{l}Consider the model of the DC motor as follows:

*νq*¨* _{rl}*=

^{Q}

^{rl}

^{e}^{+}where

*identical motors*were assumed at the LHS and RHS with the variables and parameters as follows [A. 17]:

• *Q** ^{e}*[N·

*m]*is the torque of electromagnetic origin exerted on the motor’s axle (it is proportional to the motor current),

• *Q** ^{ext}*[N·

*m]*is the torque of external origin acting on the wheel’s axle, i.e.

*Q*

_{rl}*=*

^{ext}*T*

*,*

_{rl}• *R*= 1 [Ω]is the Ohmic resistance of the motor’s coil system,

• *L*= 0.5 [H]is its inductivity,

• Θ= 0.01 [kg·*m*^{2}]denotes the momentum of the rotary part of the motor,

• *b*= 0.1 [N·*m*·*s/rad]*describes the viscous friction of the motor’s axle,

• *K*= 0.01is the motor’s torque coefﬁcient, and

• *U*[V]denotes the motor control voltage,

• *ν*= 1/5is the gear ratio regarding the motor and the wheel axles

By the use of the ﬁrst equation of the motor’s equation of motion*Q*_{rl}^{e}* ^{Des}*is calculated for

*q*¨

^{Des}*. As-suming that*

_{rl}*Q*˙

^{e}*≈0for a given constant*

_{rl}*q*˙

_{rl}*the stabilized value of the necessaryU*

_{rl}*is estimated from the 2nd equation of (6.36) as [A. 17]*

^{Des}*U*_{rl}^{Des}* ^{def}*=

*R*

*KQ*^{e}_{rl}* ^{Des}*+

*Kνq*˙

_{rl}*.*(6.37)

This approximate estimation has to be adaptively deformed.

**6.4.1.1 Simulation Results**

The above described model was used, both adaptive RFPT and RFPT based MRAC cases.

The adaptive RFPT case:

Figure 6.50: The trajectory tracking: non-adaptive control (LHS), adaptive control (RHS) [x^{N}*, y** ^{N}*: black
,θ

*: green,*

^{N}*x, y: blue,θ: pink] [A. 17]*

Figure 6.51: The tracking error (upper) and the wheels’ rotational speeds (lower): non-adaptive control
(LHS), adaptive control (RHS) [x* ^{N}*−

*x: black,y*

*−*

^{N}*y*: blue,

*θ*

*−*

^{N}*θ: green,q*˙

*: black,*

_{r}*q*˙

*: blue] [A. 17]*

_{l}The MRAC case:

Figure 6.52: The trajectories: Non Adaptive case (LHS), MRAC case (RHS) (Nominal: black, Simulated:

blue)[A. 18]

Figure 6.53: The rotation of the Cart: Non Adaptive case (LHS), MRAC case (RHS) (Nominal: green, Simulated: ocher) [A. 18]

Figure 6.54: Tracking Error: Non Adaptive case (LHS), MRAC case (RHS) (x: black, y: blue,*θ: magenta)*
[A. 18]

Figure 6.55: Rotational speed of the wheels: Non Adaptive case (LHS), MRAC case (RHS) (wheel left:

blue, wheel right: black) [A. 18]