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 ofkinematically prescribed point of the WMRfor 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, theindependent variables to be controlledare the(x, y)coordinates of a particular point of the cart on this plain, and the rotational position around the horizontal axisθ.
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 devicein whichthe rotational speeds of the wheels uniquely determine the speed of the motion over the ground.
3. For solving this task wehave only two control agents, the torques exerted by the driven wheels.
4. The dynamic model of the cart has simple equationsonly 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 areonly 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 arotation of the cart around axlexˆ1by anglequradian by the rotational matrixU(qu)generated by the generatorGUˆ as [A. 17]
U(qu)def= If the axles of the wheels in the “basic position” were parallel withxˆ1 this operation can be realized because it keeps the wheels in contact with the plane of motion that is perpendicular toxˆ3[A. 17].
Following that apply arotation aroundxˆ3with angleqvradians by the orthogonal matrixV(qv) gen-erated by the generatorGˆ(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. (qv≡θ) [A. 17].
From that fact cart is a Non-Holonomic device further constraints originate.
Since the rotationU(qu)aroundx1does not concern the position of the wheels on the ground only V(qv)is interesting that moves the wheels on the ground. (In the case of a casterU≡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=rwq˙r+ ˙ql
2 sinqv,R˙2=−rwq˙r+ ˙ql 2 cosqv
˙
qv=−rDwq˙r−2q˙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 theallowedq¨Desr andq¨Desl values that yield the best approximation of the desired 2nd derivatives if a quadratic goal function
Φ( ¨qDesr ,q¨Desl )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¨=2DIr ( ¨qr−q¨l) =Dr(Tr−Tl),
Consider the model of the DC motor as follows:
νq¨rl=Qrle+ whereidentical motorswere assumed at the LHS and RHS with the variables and parameters as follows [A. 17]:
• Qe[N·m]is the torque of electromagnetic origin exerted on the motor’s axle (it is proportional to the motor current),
• Qext[N·m]is the torque of external origin acting on the wheel’s axle, i.e.Qrlext=Trl,
• R= 1 [Ω]is the Ohmic resistance of the motor’s coil system,
• L= 0.5 [H]is its inductivity,
• Θ= 0.01 [kg·m2]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 motionQrleDesis calculated forq¨Desrl . As-suming thatQ˙erl≈0for a given constantq˙rlthe stabilized value of the necessaryUrlDesis estimated from the 2nd equation of (6.36) as [A. 17]
UrlDesdef= R
KQerlDes+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) [xN, yN: black ,θN: green,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) [xN−x: black,yN−y: blue,θN−θ: green,q˙r: black,q˙l: blue] [A. 17]
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]