Hopper Controller

Getting Started

This assumes you are running from the Lab Speedgoat PC which already has the required dependencies installed

1. Attach robot to the boom
2. Connect power and 5 pin CAN to the robot
3. Power on and index the boom
4. Ensure the legs are free to move and that upper links are horizontal
5. Turn on 48V supply to the ODrive and allow the legs to find their index points
6. Add the kinematics folder to the Matlab path
7. Open and start demo.slx on the Speedgoat
8. The green controller button will start the robot hopping and holding the red button will stop

Control Structure

Discrete state controller based on the Raibert controller ¹

  stateDiagram-v2
  direction LR
  [*] --> Rest
  Rest --> Thrust: Start
  Compression --> Rest: Stop
  Thrust --> Unloading: T1
  Unloading --> Flight: T2
  Flight --> Compression: T3
  Compression --> Thrust: T4

Loading

The controller cycles through 5 discrete states defined in States.m which sequences the control actions.

State	Control Action
Rest	Moderately stiff and damped leg to keep the body upright and standing
Thrust	Small thrust force in the $r$ direction to overcome losses and maintain a hopping. Zero leg $\phi$ axis stiffness and damping and $r$ axis damping.
Unloading	Stop applying a thrust force and let the leg retract back to its neutral point
Flight	Adjust $\phi_0$ to position foot for touchdown based on current and desired horizontal speed
Compression	Zero leg $\phi$ axis stiffness and damping, and $r$ axis damping

Transitions between controller states are triggered by changes in the robot state.

Transition	Trigger
Start	Green start button pressed on controller
T1	Leg has extended past its neutral length ( $r > r_0$ )
T2	Foot has left the ground ( $y_{foot} > 0$ )
T3	Leg is being compressed ( $\dot{r} < -1$ ) while body is falling ( $\dot{y}_{body} < 0$ )
T4	Leg is fully compressed and has begun extending ( $\dot{r} > 0$ ) while below its neutral point ( $r < r_0$ )
Stop	Red stop button pressed on controller

Leg Impedance Control

The leg is controlled in polar coordinates with a radial length $r$ and an angle $\phi$ from the body to the ankle, kind of like a pogo stick. Force in this polar frame is controlled by computing the required left $\tau_L$ and right $\tau_R$ motor torques from the desired radial force $F_r$ and torque $\tau_{\phi}$ using the leg polar Jacobian $J_p$ calculated from the leg forward kinematics.

$$ \begin{bmatrix} \tau_L \\ \tau_R \end{bmatrix} = J_p^T \begin{bmatrix} F_r \\ \tau_\phi \end{bmatrix} $$

This ignores the mass and dynamics of the legs, but is still a reasonable approximation as the links are relatively light. A PD controller on each of the polar axes enables tracking of desired position and velocity setpoints $r_0$, $\phi_0$ and $\dot{r}_0$, $\dot{\phi}_0$.

$$ F_r = K_p(r_0 - r_{leg}) + Kd_r(\dot{r_0} - \dot{r_{leg}}) + F_{thrust} $$

$$ \tau_{\phi} = Kp_{\phi}(\phi_0 - \phi_{leg}) + Kd_{\phi}(\dot{\phi_0} - \dot{\phi_{leg}}) $$

These PD controllers behave like a parallel spring and damper where $Kp$ is the spring stiffness with units $\frac{N}{m}$ or $\frac{Nm}{rad}$ and $Kd$ is the damping constant with units $\frac{Ns}{m}$ or $\frac{Nms}{rad}$. The radial stiffness determines how much the leg compresses during landing. Higher jump heights will need higher radial stiffness to stop the leg bottoming out (think pogo stick). The polar angle stiffness determines how quickly the leg repositions during flight. The damping constants are typically set to keep the axes more or less critically damped and avoid any oscillations.

Instability

NB: If the $Kp$ or $Kd$ are set too high this increases the gain crossover frequency and will cause the PD controller to go unstable. This will cause the leg to shake violently and should be avoided. Higher gains should first be tested without the legs attached. The point at which the legs go unstable is also dependent on the leg configuration and the apparent mass seen by the motors. In configurations where the Jacobian is less favourable for force production in the polar frame, high polar stiffness maps to significantly higher joint stiffnesses resulting in instability. If much higher stiffness is required in the future this can be mitigated by computing the effective joint stiffness and limiting it to a safe value.

Thrust Force

Hopping height is controlled by the radial thrust force. In an ideal world no thrust force should be required and the leg should be able to bounce up and down from an initial height on its perfectly springy leg indefinitely. Due to the motors not producing an ideal radial spring behaviour a small thrust force is required to maintain a desired hopping height. Higher thrust forces will produce greater hopping heights, but the downside to this (apart from the motors getting hotter) is that the radial leg force becomes less and less symmetric. For an ideal Raibert hopper the radial leg force is an even function of time. With a thrust force, the radial force for leg compression and extension is no longer the same producing net horizontal accelerations and impacting horizontal velocity control. For this reason the thrust force should be kept to a minimum and velocity control will most likely need to be adjusted if the thrust force is changed.

Velocity Control

The neutral point is given by

$$ x_{netral} = \frac{\dot{x}T_s}{2} $$

with $x = 0$ being directly below the hip. Placement of the foot ahead or behind the neutral point produces horizontal accelerations. A proportional controller with gain $K_v$ is used to determine foot placement and control horizontal velocity.

$$ x_{foot} = \frac{\dot{x}T_s}{2} + K_v(\dot{x} - \dot{x}_d) $$

Small changes in foot placement can produce large changes in body velocity. When tuning the gain $K_v$ start small with a value of about 0.01 which corresponds to a 1cm foot adjustment for a 1m/s body velocity error. Higher $K_v$ values will produce more repsonsive velocity control until the leg begins to overshoot. Foot placement is controlled with the $\phi$ axis of the leg by changing the setpoint $\phi_0$.

$$ \phi_0 = -\frac{\pi}{2} + \arcsin(\frac{x_{foot}}{r}) $$

Motor Encoder Indexing

The encoders in the motors are mounted to the rotor. Because of the planetary reduction, one turn of the motor output results in six turns of the rotor. There are therefore six encoder index points for a full turn of the motor output/hip. It is therefore important that the hips start close (< 60°) to the correct one of these six index points. Powering on the ODrive with the upper links horizontal should accomplish this. When powered on, the ODrive will move the right motor CCW and the left motor CW until the index pulse is detected.

Changing Feet

1. Open the kinematics folder in matlab
2. Update la.x and la.y in compute_kinematics.m with the new foot dimensions
3. Run compute_kinematics.m to update function files

Configuring ODrive

To configure the ODrive with correct motor and CAN parameters, and run motor calibration.

1. Remove the legs
2. Power ODrive with 48V
3. Connect ODrive to PC
4. Run odrive-config.py, this requires you to install odrivetool

Debugging ODrive

Simulink will not show ODrive error messages. To get useful ODrive error messages follow these steps

1. Keep simulink running - terminating will clear errors
2. Start odrivetool and connect
3. Run the dump_errors(odrv0) command and look here for the corresponding error

Dependencies

• Simulink
• Simulink Coder
• Simulink Real-Time
• Simulink Real-Time Target Support Package
• ARU Simulink Toolbox
• Speedgoat I/O Blockset

Footnotes

1. Raibert, M.H., 1986. Legged robots that balance. MIT press. ↩