Controllers

Control Structure Flow

The control algorithms on our bicycle are separated into three cascaded controllers. The output of one controller is input to another controller. In order from high level to low level:

1. Navigation controller, which tries to follow a desired path
2. Balance controller, which tries to achieve a desired lean angle (φ_d) and desired steer angle (δ_d). For straight line balance (no navigation), this controller attempts to achieve φ_d = 0 and δ_d = 0. In this position, the bicycle is balanced (upright and steered straight ahead).
3. Front motor controller, which controls front wheel angle (δ).

• Using cascaded controllers allows us to develop each controller semi-independently.

Front Motor Controller

• The front motor controller works to minimize the angle between the current steer angle (δ) and the commanded steer angle (δ_c).
• Currently (Spring 2018), the front motor controller is PD (Proportional-Derivative) controller. Specifically:

${\rm {total\_error}}=k_{p}(\delta _{c}-\delta )+k_{d}{\frac {d}{dt}}\left[\delta _{c}-\delta \right]$

• total_error is converted into a direction (DIR) and a magnitude (motor_output).

Balance Controller

• The balance controller attempts to achieve a desired lean angle (φ_d) and steer angle (δ_d). In navigation mode, the navigation algorithm sets φ_d and δ_d.
• In addition to the desired state (φ_dd, δ_d), the balance controller takes as input current lean angle (φ), lean angle rate (φ), and steer angle (δ). The balance controller outputs a steer angle rate (δ).
• Currently (Spring 2018), the balance controller is the linear controller:

$\delta _{d}^{\prime }=k_{1}(\phi -\phi _{d})+k_{2}\phi ^{\prime }+k_{3}(\delta -\delta _{d})$

Navigation Mode

• In navigation mode, φ_d ≠ 0 and δ_d ≠ 0.
• Thus, the balance controller tries to minimize the three terms in above.
• If φ − φ_d = 0 the current lean angle (φ) equals the desired lean angle (φ_d).

In other words, the bicycle has achieved the desired lean angle.

• Similarly, if δ − δ_d = 0, the bicycle has achieved the desired steer angle. Still, the bicycle should not wobble: φ'= 0.

Balance Only

• For balance only, φ_d=0, δ_d=0. The balance controller reduces to:

$\delta _{d}^{\prime }=k_{1}\phi +k<_{2}\phi ^{\prime }+k_{3}\delta$

• In balance only mode, the balance controller tries to minimize the sum of the three terms above.

1. The first term includes lean angle (φ): if φ = 0, the bike should be upright.
2. The second term includes lean angle rate (φ'): if φ'= 0, the bike should not wobble.
3. The third term includes steer angle (δ): if δ = 0, the bike should be steered straight ahead.

• If all three of these terms equal 0, the bike is balanced.

Euler Integrator

• Connects the balance controller and the front motor controller.
• Takes, as input, the steer angular rate (δ_d) commanded by the balance controller.
• Outputs a steer angle (δ_c). δ_c is input to the front motor PD controller.
• Our embedded code runs in discrete timesteps (about 0.01s each).
• δ_c is the angle the front wheel would be at in the next timestep if the wheel spins at speed ̇δ during the current timestep.
• In other words, the Euler Integrator integrates ̇δ_d to find δ_c. Specifically:

$\delta _{c}=\delta +\delta \cdot t_{step}$

• See this page for more info on Euler Integrators

Steady Lean Angle Calculation

• The balance controller takes as input a desired steer angle (δ_d) and a desired lean angle (φ_d).
• Somehow, the desired steer angle (δ_d) from the navigation algorithm must generate both a desired steer angle (δ_d) and desired lean angle

(φ_d).

• If the bicycle is in a steady turn, it must be both leaned and steered.
• Thus, to achieve a given steer angle, the bicycle must also achieve a corresponding lean angle.
• We use the equation below to calculate a lean angle (φ_d) given a steer angle (δ_d).
• This equation relates lean (φ) and steer (δ) in steady state:

$\phi _{d}={\frac {\delta _{d}v^{2}}{lg}}$