Theory about Models – AutoCarSim

The set of models: properties and methods (on short)

We propose the following succession of models (objects) for the autonomous cars (AC) and robots.

The geometrical model:
- Properties:
  - The car position;
  - The car orientation;
  - The car shape
- Methods:
  - Define the car shape in its referential frame (constructor)
  - Represent (drawing)the AC pose (position and orientation)
  - Represent the AC trajectory
  - Represent the working volume of the car which follow its trajectories
- Hypothesis:
  - No interactions between the external world and the AC; The causes are the new pose and the effects are the pose representation

The kinematical model: child of the geometrical model:
- Properties:
  - The AC initial kinematical parameters (the initial state of the AC)
  - The AC kinematical parameters:
    - Revolution speeds of wheels;
    - The steering angle and the wheel speed;
  - Sensors parameters:
    - The wheels sensor resolution;
  - Methods:
    - The AC kinematic computation: the linear and angular speed;
    - The AC pose computation (direct kinematic);
  - Hypothesis:
    - No interaction between the environment and the robot;
    - The AC kinematical parameters are the model input like commands which are not changed from intern (model) reason

Dynamic model: child of the kinematic
- Properties:
  - Inertial properties (Mass and Inertial tensor);
  - Stiffness and damping of wheels;
  - The dry and viscous friction coefficient;
- Methods:
  - Computing the AC pose (direct dynamic);
- Hypothesis:
  - The AC and environment interaction is described by the Newtonian laws.

Stochastic model child of kinematical or dynamical model:
- Properties
  - The perturbation distribution for the AC positon and orientation
- Methods
  - The perturbation convolution
- Hypothesis:
  - The AC model is kinematic or dynamic but is perturbed with a perturbation with a modeled distribution

The Differential Robot

Even if until now we haven’t seen a differential AC (and this is a pity) the differential robot (DR) is a widely used robot in industry and in robotic laboratories. We think that this is a good reason to analyses it.

The geometrical model of DR

In figure 1 an elaborate geometrical model of the DR can be seen. Shapes and color details are illustrated by this representation. In fact when we run the model and this means we simulate the geometrical model, we are interested in the robot pose (orientation and position). This means that we can use a stronger abstraction.

difrob

Figure 1. The DR geometrical model

Figure 2 presents three possible abstractions. The first is a schematic top view of the robot; the second replace the previous scheme by a triangle and the third use a referential frame to approximate the robot.

Othe repre of the DR Gm

Figure 2. Other representations of the DR geometrical model

The geometrical model properties are:

The coordinate of each point of the representation in referential frame associated with this representation;
The incident matrix of points which define lines;
The thickness of each line which is defined by the mentioned points;
The color of each line;
The incident matrix of lines which define surfaces;
The color of each surface defined by the mentioned line etc.

Figure 3 illustrate the properties of the representation proposed in figure 2b

The GM Properties

Figure 3. The geometrical model properties

$\textbf{IM}=\left[\begin{array}{cccccc} 0 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 0 & 1 \end{array}\right]\Rightarrow\begin{array}{c} L_{1}\\ L_{2}\\ L_{3}\\ L_{4}\\ L_{5} \end{array}$ (1)

The incident matrix IM has five rows (for five lines) and six columns (for six points, starting with point (x₀,y₀), and ending with point (x₅,y₅).

The GM methods refer to:

The pose computation in to an absolute referential system (where we simulate the DR locomotion). This means the possibility to represent the model for different position and orientation of the DR. So the inputs of the method can be:
- The DR referential position and orientation relative to the absolute referential frame, for an absolute pose computation (see figure 4); The outputs are the new coordinate of the GM points. The method consists on a homogenous transformation.

$\mathbf{x_{k}}=\sideset{_{0}^{k}}\mathbf{T}\mathbf{x_{0}}$

(2)

where:

$\sideset{_{0}^{k}}\mathbf{T}=\left[\begin{array}{cccc} c\alpha_{k} & -s\alpha_{k} & 0 & t_{x,k}\\ s\alpha_{k} & c\alpha_{k} & 0 & t_{y,k}\\ 0 & 0 & 1 & t_{z,k}\\ 0 & 0 & 0 & 1 \end{array}\right]}$

;

$\mathbf{x_{k}}=\left[\begin{array}{cccc} x_{0,k} & x_{1,k} & ... & x_{5,k}\\ y_{0,k} & y_{1,k} & ... & y_{5,k}\\ z_{0,k} & z_{1,k} & ... & z_{5,k}\\ 1 & 1 & ... & 1 \end{array}\right]}$

;

$\mathbf{x_{0}}=\left[\begin{array}{cccc} x_{0} & x_{1} & ... & x_{5}\\ y_{0} & y_{1} & ... & y_{5}\\ z_{0} & z_{1} & ... & z_{5}\\ 1 & 1 & ... & 1 \end{array}\right]}$

GMPoseInAbs

Figure 4. The GM poses in the absolute referential frame

The DR referential position and orientation relative to the previous pose, for a relative pose computation (see figure 5):

$\begin{displaymath} \mathbf{_{k}^{k+1}t}= \left[\begin{array}{ccc} tx_{k,k+1}& ty_{k,k+1}& tz_{k,k+1} \end{array}\right] \end{displaymath}$

$\begin{displaymath} \alpha_{k,k+1} \end{displaymath}$

GMPoseInRel

Figure 5. The GM poses in the relative referential frame

$\begin{displaymath} \mathbf{x_{k+1}}=\mathbf{_{k}^{k+1}T}\mathbf{x_{k}}= \mathbf{_{k}^{k+1}T}\mathbf{_{0}^{k}T}\mathbf{x_{0}} \end{displaymath}$

(3)

where:

$\begin{displaymath} \ {_{0}^{k}}\mathbf{T}= \left[\begin{array}{cccc} c\alpha_{k,k+1} & -s\alpha_{k,k+1} & 0 & t_{x,k,k+1}\\ s\alpha_{k,k+1} & c\alpha_{k,k+1} & 0 & t_{y,k,k+1}\\ 0 & 0 & 1 & t_{z,k}\\ 0 & 0 & 0 & 1 \end{array}\right]} \end{displaymath}$

The following observations are necessary: the chosen GM is a planar model this means that the z coordinate of all points are constant (usually zero); the orientation of the GM refers to rotations around the {Z} axis.

TrajectoryIn WV

Figure 6. Trajectory and the working volume

The GM, trajectory and working volume drawing see figure 6;
- The trajectory is defined like a succession of points defined by the origins of the referential frame of the DR’s;
- The working volume is a superposition of DR’s;

The kinematical model of DR

The hypothesis which supports this model is that the RD locomotion is determined by the left and right wheel speeds. No inertial properties of the RD will influence the phenomenon of locomotion. So wheel speeds are the cause and changing in position and in orientations are the effects.

$\begin{displaymath} v_{r},v_{l}\Rrightarrow t,\alpha \end{displaymath}$

(4)

where: are the right and left wheels speeds

In order to solve (4) we will remember another hypothesis that DR has a planar motion. Based on this assumption figure 7 illustrate the named solution.

cinematic1

Figure 7. Speed distribution for DR

$\begin{displaymath} \Bigg\{\begin{array}{cccc} v=\frac{v_r+v_l}{2}\\ \omega=\frac{v_r-v_l}{b} \end{array} \end{displaymath}$

(5)

where: ω is the angular speed of DR, b is the distance between the two wheels (see figure 6)

Additionally we can compute the radius of the gyration:

$\begin{displaymath} r=\frac{v}{\omega}=\frac{b}{2}\frac{v_r+v_l}{v_r-v_l} \end{displaymath}$

(6)

Before we derive the kinematical model an additional knowledge is important: most time the input data for the model are not the linear speed but the angular speed of the wheels. Figure 8 gives information about the link between these speeds.
Lateral View

Figure 8. Lateral view of the DR

$\begin{displaymath} v_{r,l}=R\omega_{r,l} \end{displaymath}$

(7)

And now the final step in deriving the outputs (see figure 9):

$\begin{displaymath} \Bigg\{\begin{array}{cccc} \dot{t}= \left[\begin{array}{cccc} vcos\left(\alpha\right)\\ vsin\left(\alpha\right)\\ \end{array}\right] \\ \dot{\alpha}=\omega \end{array} \end{displaymath}$

(8)

This means:

$\begin{displaymath} \Bigg\{\begin{array}{cccc} \dot{t}_{x}=\frac{v_r+v_l}{2}\cdot{cos\left(\alpha\right)}\\ \dot{t}_{y}=\frac{v_r+v_l}{2}\cdot{sin\left(\alpha\right)}\\ \omega=\frac{v_r-v_l}{b} \end{array} \end{displaymath}$

(9)

Two problems can be defined with (9). The first is the direct kinematic problem where the input (known) data are $(v_{r},v_{l})$ and the output (unknown) data are $(t_{x},t_{y},\alpha)$ . The second is the inverse kinematic problem where $(t_{x},t_{y})$ are the input data and $(v_{r},v_{l})$ are the output data.

The solution of the first problem is needed for the simulations of the DR behavior. The second problem imply the open loop hypothesis i.e. if the trajectory of the robot is impose the causes $(v_{r},v_{l})$ which produce this trajectory can be computed. Solving this problem is a nice exercise but the mentioned hypothesis is naive (because of the perturbation which can be rejected only by the close loop design).

The numerical solution of direct kinematic problem can be (10) (Euler approximation)

$\begin{displaymath} \mathbf{x}\left(k+1\right)=\mathbf{x}\left(k\right)+\left[\begin{array}{cccc} \delta_{d}\cdot{c\left(\alpha+\delta_{\alpha}\right)}\\ \delta_{d}\cdot{s\left(\alpha+\delta_{\alpha}\right)}\\ \delta_{\alpha} \end{array}\right] \end{displaymath}$

(10)

where:

$\begin{displaymath} \mathbf{x}\left(k\right)=\left[\begin{array}{cccc} t_{x}(k)\\ t_{y}(k)\\ \alpha(k) \end{array}\right] \end{displaymath}$

and

$\begin{displaymath} \Bigg\{\begin{array}{cccc} \delta_{d}=v_{k}\cdot{\Delta}t\\ \delta_{\alpha}=\omega_{k}\cdot{\Delta}t \end{array} \end{displaymath}$

DRKine

Figure 9. The DR kinematics

The numerical solutions for inverse kinematic starts from the input data (t_x,t_y) and compute the left and right wheel speeds:

$\begin{displaymath} \Bigg\{\begin{array}{cccc} \dot{t}_{x}(k)=\frac{t_{x}(k+1)-t_{x}(k)}{\Delta{t}}\\ \dot{t}_{y}(k)=\frac{t_{y}(k+1)-t_{y}(k)}{\Delta{t}} \end{array} \end{displaymath}$

(11)

$\begin{displaymath} \alpha(k)=atctg\frac{t_{y}(k+1)-t_{y}(k)}{t_{x}(k+1)-t_{x}(k)} \end{displaymath}$

(12)

$\begin{displaymath} \dot{\alpha(k)}=\frac{\alpha(k+1)-\alpha(k)}{\Delta{t}} \end{displaymath}$

(13)

$\begin{displaymath} \Bigg\{\begin{array}{cccc} v(k)=\frac{\dot{t}_{x}(k)}{cos(\alpha(k))}\\ v(k)=\frac{\dot{t}_{y}(k)}{sin(\alpha(k))} \end{array} \end{displaymath}$

(14)

$\begin{displaymath} \omega(k)=\dot{\alpha}(k) \end{displaymath}$

(15)

$\begin{displaymath} \Bigg\{\begin{array}{cccc} v_{l}(k)=\frac{v(k)-b\omega(k)}{2}\\ v_{r}(k)=\frac{v(k)+b\omega(k)}{2} \end{array} \end{displaymath}$

(16)

The acceleration of the DR has two components: the tangential acceleration and the normal acceleration.The importance of these concepts consists on illustrating the directions and the measure of the kinematic model perturbation. If the model hypothesis is accepted we will confront with perturbations like slippages in n and τ $mathbf{\thau}$ directions. Larger accelerations imply larger perturbations (slippages).

DR Acc

Figure 10. The accelerations

$\begin{displaymath} \mathbf{a}=\mathbf{a_t}+\mathbf{a_n}=\frac{dv}{dt}\tau+\frac{dv^2}{\rho}\mathbf{n} \end{displaymath}$

(17)

where: the trajectory equation is ; t is the trajectory parameter

$\begin{displaymath} \rho= \left|\frac{\left(\dot{x}^{2}+\dot{y}^{2}\right)^\frac{3}{2}}{\dot{x}(t)\ddot{y}(t)+ \dot{y}(t)\ddot{x}(t)}\right| \end{displaymath}$