5.2 Problem-Specific Elements
5.2.1 Vehicle Constraints
To understand the possible trajectories that the ATV can execute, the vehicle kinematics are first studied in the absence of dynamics. After the kinematics have been defined, the influence of dynamics on vehicle motion will be discussed.
5.2.1.1 Simple Car Kinematics
The ATV belongs to the class of vehicle described as a simple car [3]. It has four wheels that roll (without skidding) on the driving surface, of which the two rear wheels are mounted in parallel on a fixed axle and the front wheels are steered together (Figure 5.1).
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The steering angle, φ, is limited to a maximum deviation, |φ| ≤ φmax, from the vehicle
longitudinal axis. The rear wheel drive speed and the steering angle can be commanded.
The defining property for a simple car is that its wheels can rotate freely and may not skid sideways. For a fixed steering angle, this constraint is satisfied when the vehicle drives in a circle with constant radius. The geometric construction anchors the centre of the turning circle at the intersection of the extended rear and front axles and is dependent on the steering angle and the vehicle wheelbase, Lwb, such that ρ = Lwb/ tan φ. The vehicle
may also drive in a straight line (φ = 0).
The configuration space parameters for a simple car are indicated on Figure 5.1. The origin ofV is the centre of the rear axle and can be translated to (x, y) ∈ R2. The vehicle longitudinal axis can rotate with θ∈ S1, where S1is any range that signifies one revolution
e.g. [−π/2, π/2] with the end-points identified. The constraints in the configuration space can be expressed as velocity constraints over theC-space.
ρ φ θ Lw b (x, y) E N
Figure 5.1: The simple car model. The configuration parameters, q(x, y, θ), and geometric construction for the turn anchor point, as a function of φ, are indicated.
Although the simple car kinematics have been assumed for the ATV, the following prac- tical matters are noted that may degrade the model:
• The simple car model assumes that no translation is possible perpendicular to a wheel’s rolling direction, but tyres have finite stiffness that, even without wheel slip, can cause a perpendicular velocity component while it rotates.
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• The ATV has a solid rear axle without a differential drive. During a turn, the inside wheel has a shorter distance to travel than the outside wheel, but their rotational speed is equal, forcing some wheel slip. Fortunately, when driving at speed, centrifu- gal force causes some body roll and the resulting weight shift increases the frictional force of the outside wheel, concentrating wheel slip on the inner rear wheel. The simple car kinematics stay intact, based on the remaining three wheels.
• The ATV does not have a single front axle as with the simple car, but the steering mechanism has been imposed with non-linear linkage that approximately aligns the individual front axles to intersect the rear axle at the same anchor point.
• On a slippery surface, all the wheels could skid, in which case the kinematic con- straints are wholly replaced by dynamic constraints. This situation is avoided.
5.2.1.2 Dynamics
The practical vehicle has two dynamic constraints, namely finite longitudinal acceleration and finite angular slew rate on steering. Acceleration to displacement is a second order differential constraint with respect to time. A maximum speed is also set. Steering angle slew rate is a first order differential constraint with respect to time.3
5.2.1.3 Incorporating Constraints
Without dynamic constraints, the simple car can instantaneously change speed, or steering angle. This implies that a simple car can change between a sharp left-hand turn and sharp right-hand turn without any transitional phase. For the simple car, the forward speed does not alter the turning radius. The steering command solely changes the turning radius. For constant steering angles, the kinematics for the path results in simple geometric curves, i.e. arcs of a circle or straight line segments, stringed together. For smooth transitions of steering angle, the simple car is however non-holonomic and is avoided.
The steering slew rate causes smooth transitions and results in a non-holonomic car. The simple geometric curves can be regained if the vehicle is restricted to only change steering
3This paragraph only describes dominant behaviour. In fact, both constraints (acceleration and
slew rate) have higher order dynamics than stated here, but those are significantly faster and small in magnitude in comparison with the dominant behaviour. For the purpose of planning, the higher order dynamics may be safely neglected.
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angles while it is stationary [17]. Since the vehicle cannot stop instantaneously, it would have to decelerate to stop precisely at the end of the current segment, adjust its steering angle and then start the next segment. This type of movement may be acceptable while performing strictly confined operations, such as parking, but is not practical for normal driving.
To simplify planning, we use simple geometric curves, but make the following adjustments to the vehicle model:
1. The vehicle is restricted to its minimum speed, since at low speeds, the distance travelled during the transitional phase is small relative to the distance of the seg- ment. The lower the speed, the more closely the vehicle resembles a simple car.
2. During planning, a smaller value of φmax than the mechanical limit of the vehicle is
used as a safety margin. The vehicle controller is assumed to be able to correct for minor deviations from the intended path, using the safety margin as headroom to actuate.
The resulting plan will be a sub-optimal approximation, but the use of simple expressions proves useful for implementation.