📑 Formula Sheet - Control Of Mobile Robots (GeorgiaTech)

#Physics #Engineering #Control_Theory #Robotics #Math

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Module 1 - Dynamical Models and Controls

Read more: Module 1 - Introduction to Controls and Dynamical Models

Continuous Dynamical Models (Differential Equations)

Discrete Dynamical Models (Difference Equations)

Taylor Approximation:

Continuous Time PID Regulator

The control signal should be a function of the error signal "e"

Calculate Steady State error,

Discrete Time PID Regulator

Discrete P-term:
the discrete version is trivial, we just need to sample the error,

Discrete D-term
We know that the derivative of the error is roughly,

Furthermore, we can modify $K_D$ and include $\mathrm{\Delta }t$ in the calculation, we will call it $K_D^{\prime }$

Then, our controller will be,

Discrete I - term
We can approximate this area with rectangles, and now we write the sum of these rectangles as a Riemann approximation,

So after some time $\mathrm{\Delta }t$ passes, and I want to calculate my new integral, I just need to add a new rectangle to the sum,

Conclusion: To calculate the approximate integral we just need to compute,

But we can modify $K_I$ and include $\mathrm{\Delta }t$ in the calculation, we will call it $K_I^{\prime }$

Then, our controller will be,

And we just need to update "Enew" every cycle,

Discrete PID Controller

Example - pseudo-code:

Module 2 - Wheeled Robot Models

Read more: Module 2 - Introduction to Mobile Robotics and Wheeled Robot Models

Forward Kinematics: Differential Drive Model

Based on the actual physical constraints of the robot.
Inputs: Angular velocities of the wheels:

• Right wheel: ${\omega }_R$
• Left wheel: ${\omega }_L$

Forward Kinematics: Unicycle Model (Simplified Model)

Used for designing controllers.
Inputs:

• $v$: Translational (linear) velocity.
• $\omega$: Angular velocity.
  

Model Conversion (Unicycle to Differential Drive)

Geometric Formulas

Wheel Encoder Odometry

Calculating Global Obstacle Positions from Sensor Data

If the robot knows its pose ($x,y,\varphi$) and the position of obstacles in its local frame, it can compute the global position of obstacles:

Go-to-Goal Behavior: Heading-Based Proportional Controller

Closed Loop:

Key Notation:

• Robot's state:

  • Position: $(x,y)$
  • Orientation: $\theta$ (angle with respect to the $x$-axis)

• Goal's state:

  • Position: $(x_g,y_g)$

• Vector to Goal:

  • ${\overrightarrow{u}}_{gtg}$: Vector from the robot to the goal.
  • Orientation of ${\overrightarrow{u}}_{gtg}$: ${\theta }_g$ (angle with respect to the $x$-axis)

Module 3 - Linear State-Space Control Theory

Read more: Module 3 - Introduction to Linear State-Space Control Theory

General State-Space Equations (LTI System)

Steps: Converting a Linear Second-Order System to State-Space Form

Step 1: Choose State Variables & define Inputs and Outputs

Step 2: Rewrite Second-Order as a pair of First-Order Equations

Step 3: Represent in State-Space Form

The equations can be expressed as:

where:

The output equation $y=f=x_1$ can be written as:

where:

Thus, the system is represented in state-space form:

Linearizations: using a Small Angle Approximation

For small $\theta$, $\sin (\theta )\approx \theta$.

• State:$$\delta x=\left[\begin{array}{c}\delta \theta \\ \delta \dot{\theta }\end{array}\right]$$
• Dynamics (around a small $\theta$):$$\delta \dot{x}=\left[\begin{array}{cc}0& 1\\ -\frac{g}{L}& 0\end{array}\right]\delta x+\left[\begin{array}{c}0\\ c\end{array}\right]\delta u$$
• Output ($\theta$ Pendulum angle):$$\delta y=\left[\begin{array}{cc}1& 0\end{array}\right]\delta x$$

Linearization Steps: using the Jacobian Matrix

Consider a nonlinear second-order differential equation:

We need linearize it around an equilibrium point $x=0$ (meaning at $z_0=0$ and ${\dot{z}}_0=0$)

Step 1: Choose State Variables & define Inputs and Outputs

Step 2: Rewrite Second-Order as a pair of First-Order Equations

Step 3: Linearize Around an Operating Point (use the Jacobian Matrix)

At the operating point $(x_1,x_2)=(0,0)$, we use the Jacobian Matrix,

where,

• $f_1={\dot{x}}_1$
• $f_2={\dot{x}}_2$

Calculate $A$

Calculate $B$

The linearized equations become:

Step 4: Represent in State-Space Form

Now we have a Linearized system around the Operating Point: $z_o=0$, ${\dot{z}}_o=0$, $u_o=0$.

where:

and $\delta x,\delta u,\delta y$ are the difference with respect to the Operating Point.

we can expand this model like this,

Scalar Exponential and Matrix Exponential (Taylor Series)

Time derivative of the Exponential Function

Derivative of the Exponential:

Time derivative of the scalar exponential:

Time derivative of the matrix exponential (behaves like a scalar exponential):

Matrix Diagonalization

Let’s say there exists an invertible matrix $P$ such that:

where:

• $A_{\text{ugly}}$: The original "ugly" matrix.
• $P$: The matrix with the change of basis (constructed from the eigenvectors of $A_{\text{ugly}}$).
• $D$: A "pretty" diagonal matrix containing the eigenvalues of $A_{\text{ugly}}$.

By induction, we can generalize:

Matrix Exponential Diagonalization

Applying this to the matrix exponential as $e^{Dt}$, we have:

where:

and:

where each column $v_i$ is the eigenvector corresponding to eigenvalue ${\lambda }_i$.

If $D$ is a diagonal matrix, its exponential is straightforward:

Thus, the final result is:

Eigenvalues (Characteristic Equation)

Eigenvectors

The Null Space of a Linear Transformation

The null space of $A$ is defined as:

It means, that the null space of $A$ is the set of all vectors $x$ that satisfy $Ax=0$

Solution to the Homogeneous First-Order Linear Differential Equation

This is a first-order linear differential equation, it is a scalar system, where $x$ is a single number, not a vector:

The solution to this equation is:

where:

• $x_0$ is the initial condition at $t_0$.
• $e^{a(t-t_0)}$ is the scalar exponential.

Stability of the Scalar System

Solution to the System of Coupled Homogeneous First-Order Linear Differential Equations (Matrix Case)

For systems where $x\in {\mathbb{R}}^n$, the dynamics are:

The solution remains similar to the scalar version, but uses a matrix exponential:

Here, $e^{A(t-t_0)}$ is the matrix exponential, defined as a Taylor Series:

The State Transition Matrix

it can be used to express the general general solution,

The properties of the matrix exponential still apply,

General Solution for a Controlled System

For systems with inputs:

The solution incorporates the input's influence:

Key Terms:

• $\mathrm{\Phi }(t,t_0)$: The state transition matrix.
• ${\int }_{t_0}^t\mathrm{\Phi }(t,\tau )Bu(\tau )d\tau$: The effect of the input $u(\tau )$ on the state over time. Also known as a Convolution Integral.

For systems with measured outputs:

The output is derived directly from the state solution.

Note: If $A$ is diagonalizable, then: $\mathrm{\Phi }(t,t_0)=P{e}^{D(t-t_0)}{P}^{-1}$

Stability of Higher Dimension Systems

The Rendezvous Problem in Swarm Robotics

• Robots share relative positions.
• Consensus equations define intermediary goal points.
  

the system can be written as:

where $L$ is the Graph Laplacian Matrix, describing the communication structure (who talks to whom).

Graph Laplacian Properties:

1. $L$ has:
   • One zero eigenvalue (${\lambda }_1=0$) if the graph is connected.
   • All other eigenvalues are positive (${\lambda }_2,{\lambda }_3,\cdots >0$).
2. $-L$ has:
   • One zero eigenvalue (${\lambda }_1=0$).
   • All other eigenvalues are negative (${\lambda }_2,{\lambda }_3,\cdots <0$).
   • Therefore the multiple-robot case has Critical Stability

The null space of $L$ determines the consensus value. If the graph is connected:

Module 4 - Linear State-Space Control Design

Read more: Module 4 - Linear State-Space Control Design