Controllers

Control Structure Flow

Intro

• 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:

δd' = k1φ + k2φ' + k3δ

• 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 ̇δd to find δc. Specifically:

δc = δ + δ·tstep

• 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, t oachieve 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:

φd = δdv2/(lg)