Wobbly Robot Model

The wobbly robot model simulates a two-wheeled differential-drive mobile robot with a low center of gravity. The body is designed with enough ground clearance to tilt, but the low center of mass will wobble it back upright. The body includes a mast with two distance sensors, a radio receiver and emitter, GPS receiver for location, compass for heading, and three-axis accelerometer. Each wheel has both a rotational motor and a position sensor.

The shape is constructed entirely of primitive cylinders.The same geometry is referenced to use as bounding objects for collision and automatic calculation of physics parameters.

The user-accessible parameters include the wheel radius, axle length, body color, and wheel color. The kinematics, shape, and physics will parametrically scale, but the actuator parameters do not change.

This model is demonstrated in the wobbly-demo.wbt world.

Screenshot of Webots model of a diff-drive ‘wobbly’ robot with low center of gravity.

Contents

• Wobbly Robot Model

  • System Kinematics and Components

  • wobbly.proto

  • Sample Control Code

System Kinematics and Components

The bodies are as follows:

name	color	notes
body	blue	the core solid including counterweight, stalk, axle, and sensor disc
left wheel	red	left wheel disc, cylinder with parameterized radius
right wheel	red	right wheel disc, cylinder with parameterized radius

The joints are as follows:

name	parent	child	notes
(6DOF free)	world	body	mobile body; body +X axis is ‘forward’, +Z is ‘up’
left wheel	body	left wheel	rotational joint on axle, parameterized Z position
right wheel	body	right wheel	rotational joint on axle, parameterized Z position

The joint axes are as follows:

name	direction	notes
left wheel	along +Y	along axle; direction chosen so positive rotation moves forward
right wheel	along +Y	along axle; direction chosen so positive rotation moves forward

The motors and sensors are named as follows:

name	notes
left wheel motor	RotationalMotor on left wheel
right wheel motor	RotationalMotor on right wheel
left wheel sensor	PositionSensor on left wheel
right wheel sensor	PositionSensor on right wheel
leftDistanceSensor	DistanceSensor near top of body, aimed left of enter
rightDistanceSensor	DistanceSensor near top of body, aimed right of center
gps	GPS locator mounted on body in center of axle
compass	compass mounted on body in center of axle
accelerometer	three-axis accelerometer mounted on body in center of axle
receiver	radio Receiver
emitter	radio Emitter

wobbly.proto

The robot model has been encapsulated in a .proto file for easy reuse. The model includes user-accessible scaling and color parameters.

#VRML_SIM R2021b utf8
# Two-wheeled differential-drive mobile robot with a low center of gravity.  The
# body is designed with enough ground clearance to tilt, but the low center of
# mass will wobble it back upright.  The body includes a mast with two distance
# sensors, a radio receiver and emitter, and a GPS receiver.  Each wheel has
# both a rotational motor and a position sensor. The shape is constructed
# entirely of primitive cylinders.
# documentation url: https://courses.ideate.cmu.edu/16-375
# license: No copyright, 2020-2021 Garth Zeglin.  This file is explicitly placed in the public domain.
PROTO wobbly [
  field SFVec3f    translation  0 0 0
  field SFRotation rotation     0 1 0 0
  field SFString   controller   "wobbly"
  field SFString   name         "Wobbly"
  field SFFloat    wheelRadius  0.1
  field SFFloat    axleLength   0.14
  field SFColor    bodyColor    0.0820075 0.364731 0.8
  field SFColor    wheelColor   1 0 0
  field SFString   customData   ""
]
{
  Robot {
    # connect properties to user-visible data fields
    translation IS translation
    rotation IS rotation
    controller IS controller
    name IS name
    customData IS customData

    # calculate derived parameters
    %{
      local counterweightHeight = fields.wheelRadius.value
      local counterweightOffset = 0.75*fields.wheelRadius.value
      local counterweightTop    = 1.25*fields.wheelRadius.value
      local sensorHeight        = 3*fields.wheelRadius.value
      local stalkHeight         = sensorHeight - counterweightTop
      local stalkOffset         = counterweightTop + 0.5*stalkHeight
    }%

    children [
      # The body group contains three cylinders: the massive counterweight at
      # bottom, a thin stalk rising above it, topped by the sensor disc.
      # All these shapes participate in collision; the axle is kept separate
      # as it is just for rendering.
      DEF bodyShape Group {
        children [
          # counterweight
          Transform {
            translation 0 0 %{= counterweightOffset }%
            rotation 1 0 0 1.5708
            children [
              Shape {
                appearance DEF bodyAppearance PBRAppearance {
                  baseColor IS bodyColor
                  roughness 0.5
                  metalness 0.5
                }

                geometry Cylinder {
                  height %{= counterweightHeight }%
                  radius 0.05
                }
              }
            ]
          }
  	  # sensor disc
          Transform {
            translation 0 0 %{= sensorHeight }%
            rotation 1 0 0 1.5708
            children [
              Shape {
                appearance USE bodyAppearance
                geometry Cylinder {
                  height 0.01
                  radius 0.025
                }
              }
            ]
          }
	  # stalk
          Transform {
            translation 0 0 %{= stalkOffset }%
            rotation 1 0 0 1.5708
            children [
              Shape {
                appearance USE bodyAppearance
                geometry Cylinder {
                  height %{= stalkHeight }%
                  radius 0.01
                }
              }
            ]
          }
        ]
      }
      # Visible axle shape, not part of the boundingObject.
      DEF axleShape Transform {
        translation 0 0 %{= fields.wheelRadius.value }%
        children [
          Shape {
            appearance USE bodyAppearance
            geometry Cylinder {
              height %{= fields.axleLength.value + 0.02 }%
              radius 0.005
            }
          }
        ]
      }
      # Define the left wheel axis, pointing in the +Y direction along the axle.
      HingeJoint {
        jointParameters HingeJointParameters {
          axis 0 1 0
          anchor 0 0 %{= fields.wheelRadius.value }%
        }
        device [
          RotationalMotor {
            name "left wheel motor"
            acceleration 10
            maxTorque 2
          }
          PositionSensor {
            name "left wheel sensor"
          }
        ]
	# Define the left wheel solid, offset along +Y along the axle.
        endPoint DEF leftWheel Solid {
          translation 0 %{= 0.5*fields.axleLength.value }% %{= fields.wheelRadius.value }%
          rotation 0 -1 0 0
          children [
            # Define the left wheel shape and appearance, which is used by the right wheel solid.
            DEF wheelShape Transform {
              rotation 0 0 1 0
              children [
                Shape {
                  appearance PBRAppearance {
                    baseColor IS wheelColor
                    roughness 0.5
                    metalness 0.5
                  }
                  geometry Cylinder {
                    height 0.01
                    radius %{= fields.wheelRadius.value }%
                  }
                }
              ]
            }
          ]
          name "left wheel"
          boundingObject USE wheelShape
          physics DEF wheelPhysics Physics {
            density 600
          }
        }
      }
      # Define the right wheel axis, also pointing in the +Y direction along the axle
      # so positive wheel rotations will move forward.
      HingeJoint {
        jointParameters HingeJointParameters {
          axis 0 1 0
          anchor 0 0 %{= fields.wheelRadius.value }%
        }
        device [
          RotationalMotor {
            name "right wheel motor"
            acceleration 10
            maxTorque 2
          }
          PositionSensor {
            name "right wheel sensor"
          }
        ]
	# Define the right wheel solid, offset along -Y along the axle.
	# The wheel shape is inherited from the left wheel.
        endPoint DEF rightWheel Solid {
          translation 0 %{= -0.5*fields.axleLength.value }% %{= fields.wheelRadius.value }%
          rotation 0 0 1 3.14159
          children [
            USE wheelShape
          ]
          name "right wheel"
          boundingObject USE wheelShape
          physics USE wheelPhysics
        }
      }
      # Define the two range sensors.  They are located on the sensor disc on
      # top of the body stalk, each pointed off-axis by 1.0 radian.
      DEF leftEyeSensor DistanceSensor {
        translation 0.019 0.016 %{= sensorHeight }%
        rotation 0 0 1 0.7
        children [
          DEF eyeShape Transform {
            rotation 0 0 1 1.57
            children [
              Shape {
                appearance PBRAppearance {
                  baseColor 0.975691 0.981481 0.0252992
                  roughness 0.5
                  metalness 0.5
                }
                geometry Cylinder {
                  height 0.004
                  radius 0.005
                }
              }
            ]
          }
        ]
        name "leftDistanceSensor"
        lookupTable [
          0 0.025 0
          1 1 0
        ]
        numberOfRays 2
        aperture 0.4
      }
      DEF rightEyeSensor DistanceSensor {
        translation 0.019 -0.016 %{= sensorHeight }%
        rotation 0 0 1 -0.7
        children [
          USE eyeShape
        ]
        name "rightDistanceSensor"
        lookupTable [
          0 0.025 0
          1 1 0
        ]
        numberOfRays 2
        aperture 0.4
      }
      # add a default radio receiver and transmitter
      Receiver {
      }
      Emitter {
      }
      # add a position-sensing 'GPS' device located at the center of the axle
      GPS {
        translation 0 0 %{= fields.wheelRadius.value }%
      }
      # add a rotation-sensing 'compass' returning only the X and Y components of the North vector
      Compass {
        zAxis FALSE
      }
      # add an accelerometer located at the center of the axle to measure both gravity and reactive forces
      Accelerometer {
        translation 0 0 %{= fields.wheelRadius.value }%
      }

    ] # close children list of Robot
    boundingObject USE bodyShape
    physics Physics {
      density -1
      mass 3
    }
  } # close Robot
}

Sample Control Code

# wobbly.py

# Sample Webots controller file for driving the wobbly diff-drive
# mobile robot.

# No copyright, 2020-2021, Garth Zeglin.  This file is
# explicitly placed in the public domain.

print("loading wobbly.py...")

# Import the Webots simulator API.
from controller import Robot

# Import standard Python libraries.
import math, random, time

# Define the time step in milliseconds between controller updates.
EVENT_LOOP_DT = 200

################################################################
class Wobbly(Robot):
    def __init__(self):

        super(Wobbly, self).__init__()
        self.robot_name = self.getName()
        print("%s: controller connected." % (self.robot_name))

        # Attempt to randomize the random library sequence.
        random.seed(time.time())

        # Initialize geometric constants.  These should match
        # the current geometry of the robot.
        self.wheel_radius = 0.1
        self.axle_length  = 0.14

        # Fetch handles for the wheel motors
        self.l_motor = self.getDevice('left wheel motor')
        self.r_motor = self.getDevice('right wheel motor')

        # Adjust the low-level controller gains.
        print("%s: setting PID gains." % (self.robot_name))
        self.l_motor.setControlPID(1.0, 0.0, 0.1)
        self.r_motor.setControlPID(1.0, 0.0, 0.1)

        # Fetch handles for the wheel joint sensors.
        self.l_pos_sensor = self.getDevice('left wheel sensor')
        self.r_pos_sensor = self.getDevice('right wheel sensor')

        # Specify the sampling rate for the joint sensors.
        self.l_pos_sensor.enable(EVENT_LOOP_DT)
        self.r_pos_sensor.enable(EVENT_LOOP_DT)

        # Connect to the eye sensors.
        self.l_eye_sensor = self.getDevice('leftDistanceSensor')
        self.r_eye_sensor = self.getDevice('rightDistanceSensor')
        self.l_eye_sensor.enable(EVENT_LOOP_DT)
        self.r_eye_sensor.enable(EVENT_LOOP_DT)

        # Connect to the radio emitter and receiver.
        self.receiver = self.getDevice('receiver')
        self.emitter  = self.getDevice('emitter')
        self.radio_interval = 1000
        self.radio_timer = 0
        self.receiver.enable(self.radio_interval)

        # Maintain a table of peer robot locations received over the radio.
        self.peers = {}

        # Connect to the GPS position sensor.
        self.gps = self.getDevice('gps')
        self.gps_timer = 0
        self.gps_interval = 1000
        self.gps.enable(self.gps_interval)
        self.gps_location = [0, 0, 0.1] # reference pose value

        # Connect to the compass orientation sensor.
        self.compass = self.getDevice('compass')
        self.compass_timer = 0
        self.compass_interval = 200
        self.compass.enable(self.compass_interval)

        # State variables for reporting compass readings.  The heading is the
        # body forward (body +X) direction expressed in degrees clockwise from
        # North (world +Y).  The compass_vector is the direction of North
        # expressed in body coordinates.  The neutral pose points East.
        self.heading = 90
        self.compass_vector = [0,1] # reference pose value
        self.heading_error = 0

        # Connect to the accelerometer sensor.
        self.accelerometer = self.getDevice('accelerometer')
        self.accelerometer_timer = 0
        self.accelerometer_interval = 200
        self.accelerometer.enable(self.accelerometer_interval)
        self.accel_vector = [0, 0, 9.81] # reference pose value

        # Initialize generic behavior state machine variables.
        self.state_timer = 0        # timers in milliseconds
        self.state_index = 0        # current state
        self.target_heading = 0
        self.target_velocity = 0
        return

    #================================================================
    # Polling function to process sensor input at different timescales.
    def poll_sensors(self):

        self.gps_timer -= EVENT_LOOP_DT
        if self.gps_timer < 0:
            self.gps_timer += self.gps_interval
            location = self.gps.getValues()
            if not math.isnan(location[0]):
                self.gps_location = location
                # print("%s GPS: %s" % (self.robot_name, location))

        self.compass_timer -= EVENT_LOOP_DT
        if self.compass_timer < 0:
            self.compass_timer += self.compass_interval
            orientation = self.compass.getValues()
            if not math.isnan(orientation[0]) and not math.isnan(orientation[1]):
                # For heading, 0 degrees is North, 90 is East, 180 is South, 270 is West.
                # The world is assumed configured 'ENU' so X is East and Y is North.
                # The robot 'front' is along body +X, so the neutral pose is facing East.
                self.heading = math.fmod(2*math.pi + math.atan2(orientation[1], orientation[0]), 2*math.pi) * (180.0/math.pi)
                self.compass_vector = orientation[0:2]
                # print("%s Compass: %s, heading %3.0f deg" % (self.robot_name, self.compass_vector, heading))

        self.accelerometer_timer -= EVENT_LOOP_DT
        if self.accelerometer_timer < 0:
            self.accelerometer_timer += self.accelerometer_interval
            self.accel_vector = self.accelerometer.getValues()
            # The accelerometer will read [0, 0, 9.81] when stationary in the reference pose.
            # print("%s Accelerometer: %s" % (self.robot_name, self.accel_vector))

        return

    #================================================================
    # Polling function to process radio and network input at different timescales.
    def poll_communication(self):
        self.radio_timer -= EVENT_LOOP_DT
        if self.radio_timer < 0:
            self.radio_timer += self.radio_interval
            while self.receiver.getQueueLength() > 0:
                packet = self.receiver.getData()
                # print("%s Receiver: %s" % (self.robot_name, packet))
                tokens = packet.split()
                if len(tokens) != 5:
                    print("%s malformed packet: %s" % (self.robot_name, packet))
                else:
                    name = tokens[0].decode() # convert bytestring to Unicode
                    if self.peers.get(name) is None:
                        print("%s receiver: new peer observed: %s" % (self.robot_name, name))

                    self.peers[name] = {'location' : [float(tokens[1]), float(tokens[2]), float(tokens[3])],
                                        'heading'  : float(tokens[4]),
                                        'timestamp' : self.getTime(),
                                        }

                # done with packet processing
                self.receiver.nextPacket()

            # Transmit a status message at the same rate
            name_token = self.robot_name.replace(" ","_")
            status = "%s %.2f %.2f %.2f %.0f" % (name_token, self.gps_location[0], self.gps_location[1],
                                                 self.gps_location[2] - self.wheel_radius, self.heading)

            # emitter requires a bytestring, not a Python Unicode string
            data = status.encode()

            # print("%s emitter: sending %s" % (self.robot_name, data))
            self.emitter.send(data)

    #================================================================
    # motion primitives
    def go_forward(self, velocity):
        """Command the motor to turn at the rate which produce the ground velocity
           specified in meters/sec.  Negative values turn backward. """

        # velocity control mode
        self.l_motor.setPosition(math.inf)
        self.r_motor.setPosition(math.inf)

        # calculate the rotational rate in radians/sec based on the wheel radius
        theta_dot = velocity / self.wheel_radius
        self.l_motor.setVelocity(theta_dot)
        self.r_motor.setVelocity(theta_dot)
        return

    def go_rotate(self, rot_velocity):
        """Command the motors to turn in place at the rate which produce the rotational
           velocity specified in radians/sec.  Negative values turn
           backward."""

        # velocity control mode
        self.l_motor.setPosition(math.inf)
        self.r_motor.setPosition(math.inf)

        # calculate the difference in linear velocity of the wheels
        linear_velocity = self.axle_length * rot_velocity

        # calculate the net rotational rate in radians/sec based on the wheel radius
        theta_dot = linear_velocity / self.wheel_radius

        # apply the result symmetrically to the wheels
        self.l_motor.setVelocity( 0.5*theta_dot)
        self.r_motor.setVelocity(-0.5*theta_dot)
        return

    def heading_difference(self, target, current):
        """Calculate a directional error in degrees, always returning a value on (-180, 180]."""
        err = target - current
        # fold the range of values to (-180, 180]
        if err > 180.0:
            return err - 360
        elif err <= -180.0:
            return err + 360
        else:
            return err

    def go_heading(self, target_heading):
        """Rotate toward the heading specifed in positive degrees, with 0 at North (+Y),
           90 at East (+X).  This assume the compass-reading process is
           active."""

        # find the directional error in degrees
        self.heading_error = self.heading_difference(target_heading, self.heading)

        # apply a linear mapping from degrees error to rotational velocity in radians/sec
        rot_vel = 0.02 * self.heading_error
        self.go_rotate(rot_vel)
        # print("go_heading: %f, %f, %f" % (target_heading, self.heading, rot_vel))
        return

    def go_still(self):
        """Actively damp any wobble to come to rest in place."""
        # map an error in X acceleration to a linear velocity
        vel = -0.05 * self.accel_vector[0]
        self.go_forward(vel)
        # print("go_still: %f, %f" % (self.accel_vector[0], vel))
        return

    #================================================================
    def peer_heading_distance(self, record):
        """Given a peer record, return a tuple (heading, distance) with the compass
           heading and distance in meters of the peer from this robot current
           location."""
        loc = record['location']
        dx = loc[0] - self.gps_location[0]
        dy = loc[1] - self.gps_location[1]
        distance = math.sqrt(dx*dx + dy*dy)
        heading  = math.fmod(2*math.pi + math.atan2(dx, dy), 2*math.pi) * (180.0/math.pi)
        return heading, distance

    def nearest_peer(self, range=2.0):
        """Locate the nearest peer (as reported by radio) within the given range.
           Returns either None or a dictionary with the location record."""
        result = None
        best = math.inf
        for name in self.peers:
            record = self.peers[name]
            heading, dist = self.peer_heading_distance(record)
            if dist < best:
                result = record
                best = dist
        return result

    #================================================================
    def poll_wandering_activity(self):
        """State machine update function to aimlessly wander around the world."""

        # This machine always transitions at regular intervals.
        timer_expired = False
        if self.state_timer < 0:
            self.state_timer += 3000
            timer_expired = True

        # Evaluate the side-effects and transition rules for each state.
        if self.state_index == 0:
            print("Init state, entering cycle.")
            self.state_index = 1

        elif self.state_index == 1:
            self.go_forward(0.2)

            if timer_expired:
                self.state_index += 1

        elif self.state_index == 2:
            self.go_heading(self.target_heading)

            if timer_expired:
                self.state_index += 1

        elif self.state_index == 3:
            self.go_still()

            if timer_expired:
                self.state_index += 1

        elif self.state_index == 4:
            self.go_rotate(math.pi / 6)

            if timer_expired:
                self.state_index = 1
                self.target_heading = random.randint(0,360)

        else:
            print("%s: invalid state, resetting." % (self.robot_name))
            self.state_index = 0

        if timer_expired:
            print("%s: transitioning to state %s" % (self.robot_name, self.state_index))

        return

    #================================================================
    def poll_following_activity(self):
        """State machine update function to always move toward the nearest peer."""

        if self.state_timer < 0:
            self.state_timer += 1000

            # periodically test if there is a nearby peer
            nearest = self.nearest_peer()
            if nearest is None:
                self.state_index = 1
            else:
                self.state_index = 2
                heading, distance = self.peer_heading_distance(nearest)
                self.target_heading = heading
                self.target_velocity = 0.1 * distance
                print("%s: peer to follow at %f deg, %f meters" % (self.robot_name, heading, distance))

        # always either stabilize, turn, or move
        if self.state_index < 1:
            self.go_still()

        else:
            heading_err = self.heading_difference(self.target_heading, self.heading)
            if abs(heading_err) > 20.0 or abs(self.target_velocity) < 0.05:
                self.go_heading(self.target_heading)
            else:
                self.go_forward(self.target_velocity)

    #================================================================
    def run(self):
        # Run loop to execute a periodic script until the simulation quits.
        # If the controller returns -1, the simulator is quitting.
        while self.step(EVENT_LOOP_DT) != -1:
            # Read simulator clock time.
            self.sim_time = self.getTime()

            # Read sensor values.
            self.poll_sensors()

            # Check the radio and/or network.
            self.poll_communication()

            # Update the activity state machine.
            self.state_timer -= EVENT_LOOP_DT

            # This will run some open-loop motion.  One robot will be the leader, the rest will follow.
            mode = self.getCustomData()
            if mode == 'leader':
                self.poll_wandering_activity()
            else:
                self.poll_following_activity()


################################################################
# Start the script.
robot = Wobbly()
robot.run()