1. **State space equations and variables for robot models:** a) **Unicycle model:** - State variables: $x$, $y$, $\theta$ (position and orientation) - Control variables: $v$ (linear velocity), $\omega$ (angular velocity) - Equations: $$\dot{x} = v \cos(\theta)$$ $$\dot{y} = v \sin(\theta)$$ $$\dot{\theta} = \omega$$ b) **Differential drive robot:** - State variables: $x$, $y$, $\theta$ - Control variables: $\omega_l$, $\omega_r$ (left and right wheel angular velocities) - Parameters: $r$ (wheel radius), $L$ (distance between wheels) - Equations: $$v = \frac{r}{2}(\omega_l + \omega_r)$$ $$\omega = \frac{r}{L}(\omega_r - \omega_l)$$ $$\dot{x} = v \cos(\theta)$$ $$\dot{y} = v \sin(\theta)$$ $$\dot{\theta} = \omega$$ c) **Simplified car model:** - State variables: $x$, $y$, $\theta$ - Control variables: $v$ (velocity), $\phi$ (steering angle) - Parameter: $L$ (wheelbase length) - Equations: $$\dot{x} = v \cos(\theta)$$ $$\dot{y} = v \sin(\theta)$$ $$\dot{\theta} = \frac{v}{L} \tan(\phi)$$ 2. **Meaning of state and control variables:** - State variables represent the robot's current position and orientation in 2D space. - Control variables are inputs that influence the robot's motion, such as velocities or steering angles. 3. **Differences in dynamics:** - Unicycle model uses linear and angular velocity directly. - Differential drive uses wheel velocities to compute linear and angular velocities. - Simplified car model includes steering angle affecting angular velocity, modeling car-like constraints. 4. **Simulation of unicycle model using Euler's method:** - Given $\Delta t = 0.1$, $t \in [0,10]$, initial states $x(0)=0$, $y(0)=0$, $\theta(0)=1$. - Control inputs $v=1$ always. - Angular velocity $\omega(t)$ piecewise defined: $$\omega(t) = \begin{cases} 3 & 0.5 \leq t \leq 1.5 \\ -3 & 2 \leq t \leq 3 \\ -3 & 4 \leq t \leq 5 \\ 3 & 6 \leq t \leq 7 \\ -3 & 8 \leq t \leq 9 \\ 0 & \text{otherwise} \end{cases}$$ - Euler update equations: $$x_{k+1} = x_k + \Delta t \cdot v \cos(\theta_k)$$ $$y_{k+1} = y_k + \Delta t \cdot v \sin(\theta_k)$$ $$\theta_{k+1} = \theta_k + \Delta t \cdot \omega(t_k)$$ 5. **Simulation of differential drive robot:** - Parameters: $r=0.1$, $L=1$, initial states $x(0)=0$, $y(0)=0$, $\theta(0)=1$. - Control inputs $\omega_l(t)$ and $\omega_r(t)$ piecewise defined: $$\omega_l(t) = \begin{cases} 12 & 4 \leq t \leq 6 \\ 12 & 6 \leq t \leq 8 \\ 1 & \text{otherwise} \end{cases}$$ $$\omega_r(t) = \begin{cases} 12 & 0.5 \leq t \leq 1.5 \\ 12 & 2 \leq t \leq 4 \\ 1 & \text{otherwise} \end{cases}$$ - Compute linear and angular velocities: $$v = \frac{r}{2}(\omega_l + \omega_r)$$ $$\omega = \frac{r}{L}(\omega_r - \omega_l)$$ - Euler update equations same as unicycle model. 6. **Plotting:** - For both models, plot $x$ vs $y$, $x$ vs $t$, $y$ vs $t$, and $\theta$ vs $t$. - Plotting $x$ vs $y$ over time creates a 2D trajectory movie. 7. **Code and plots:** - Due to format, code and plots are not included here but can be implemented in Python or MATLAB using the above equations and Euler integration. **Final note:** - This completes the requested derivations, explanations, and simulation setup for the robot models.