Pendulum 1-2 Robot Model

This sample project includes a model of a underactuated robot structured as a double pendulum with two distal links. The proximal link is actuated with a torque motor, and the distal links use passive pivots with only friction.

This model is demonstrated in the pendulum-1-2-demo.wbt world.

Screenshot of Webots model of 1-2 pendulum with a driven base joint and two passive distal joints. The robot is shown in the neutral pose.

System Kinematics

The bodies are as follows:

name	color	notes
base	blue	massive base resting on ground
link1	red	pivoting arm, with center of mass offset above the joint axis
link2A	green	upper passively pivoting arm,
link2B	green	lower passively pivoting arm, somewhat shorter than link2A

The joints are as follows:

name	parent	child	notes
joint1	base	link1	includes motor1 and the joint1 sensor
joint2A	link1	link2A	includes joint2A position sensor
joint2B	link1	link2A	includes joint2B position sensor

The axes are as follows:

name	direction	notes
joint1	along X	located 1.0 meters above the ground plane
joint2A	along X	located 0.5 meters about the joint1 axis
joint2B	along X	located 0.3 meters below the joint1 axis

Design Notes

1. The robot is directly modeled in the scene tree so all parameters can be visible. In general it is more flexible to transfer models to .proto files so they can be instantiated more than once.

2. For simplicity, the model uses only primitive shapes (boxes). The same shapes are used for both rendering and contact detection (i.e. bounding objects). This also allows the simulator to automatically calculate inertia properties.

3. Each Solid node uses the translation vector to place the link body coordinate origin at a logical point on the joint axis. This convention is intended to keep the hinge vectors more legible.

4. Each Shape node has been wrapped in a Transform node to ease positioning it with the link body coordinates.

5. The sample controller runs an event loop which calculates joint1 torques based on elapsed time. It could be extended to use joint position data for feedback.

CAD Drawings

The following images were generated using a Fusion 360 CAD model independently drawn to the same dimensions as the Webots robot model.

Dimensioned sketch of CAD model of 1-2 pendulum, front view, showing all X-Z dimensions.

CAD model of 1-2 pendulum.

CAD model of 1-2 pendulum, front and right views.

Sample Robot Control Code

# pendulum_1_2.py
#
# Sample Webots controller file for driving the
# underactuated 1-2 pendulum.  The robot has a driven
# base joint between the base and link1, and passive
# joints between link1 and the two distal links link2A
# and link2B.
#
# No copyright, 2020-2021, Garth Zeglin.  This file is
# explicitly placed in the public domain.

print("loading pendulum_1_2.py...")

# Import the Webots simulator API.
from controller import Robot

# Import the standard Python math library.
import math

print("pendulum_1_2.py waking up...")

# Define the time step in milliseconds between
# controller updates.
EVENT_LOOP_DT = 20

# Request a proxy object representing the robot to
# control.
robot = Robot()

# Fetch handles for the joint sensors.
j1  = robot.getDevice('joint1')
j2A = robot.getDevice('joint2A')
j2B = robot.getDevice('joint2B')

# Specify the sampling rate for the joint sensors.
j1.enable(EVENT_LOOP_DT)
j2A.enable(EVENT_LOOP_DT)
j2B.enable(EVENT_LOOP_DT)

# Fetch handle for the 'base' joint motor.  In this
# example the motor will be controlled as a torque
# motor, bypassing the lower-level PID control.
motor1 = robot.getDevice('motor1')
motor1.setTorque(0.0)

# Run an event loop until the simulation quits,
# indicated by the step function returning -1.
while robot.step(EVENT_LOOP_DT) != -1:

    # Read simulator clock time.
    t = robot.getTime()

    # Read the new joint positions.
    q1  = j1.getValue()
    q2A = j2A.getValue()
    q2B = j2B.getValue()

    # Compute and apply new base joint actuator torque.
    # In this example, the excitation is only based on
    # time, but could also be a function of the joint
    # positions.
    tau = 3 * math.sin(3*t)
    motor1.setTorque(tau)

World File

#VRML_SIM R2023b utf8
EXTERNPROTO "https://raw.githubusercontent.com/cyberbotics/webots/R2023b/projects/objects/floors/protos/RectangleArena.proto"
EXTERNPROTO "https://raw.githubusercontent.com/cyberbotics/webots/R2023b/projects/appearances/protos/PaintedWood.proto"

WorldInfo {
}
Viewpoint {
  orientation 0.20637375705407857 0.11296232240268078 -0.9719307517085652 2.083640805255243
  position 1.7870404774717816 3.2844836209794273 2.3418665277932647
}
Background {
  skyColor [
    0.1 0.1 0.1
  ]
}
DirectionalLight {
  direction 0.4 -0.5 -1
  intensity 3
  castShadows TRUE
}
RectangleArena {
  rotation 1.8366025517039032e-06 -1.836602551703903e-06 0.9999999999966269 1.5707963267982696
  floorSize 2 2
}
Robot {
  rotation 0 0 1 1.5708
  children [
    DEF baseObject Transform {
      translation 0 0 0.6
      children [
        Shape {
          appearance PaintedWood {
            colorOverride 0.21529 0.543008 0.99855
          }
          geometry Box {
            size 0.3 0.5 1.2
          }
        }
      ]
    }
    HingeJoint {
      jointParameters HingeJointParameters {
        anchor 0 0 1
      }
      device [
        PositionSensor {
          name "joint1"
        }
        RotationalMotor {
          name "motor1"
          acceleration 2
          maxVelocity 3.14
          minPosition -10
          maxPosition 10
          maxTorque 20
        }
      ]
      endPoint Solid {
        translation 0.15 0 1
        rotation 1 0 0 0
        children [
          DEF link1Shape Transform {
            translation 0.08 0 0.1
            children [
              Shape {
                appearance PaintedWood {
                  colorOverride 0.990494 0.516915 0.468254
                }
                geometry Box {
                  size 0.1 0.3 1
                }
              }
            ]
          }
          HingeJoint {
            jointParameters HingeJointParameters {
              anchor 0 0 0.5
              dampingConstant 0.1
            }
            device [
              PositionSensor {
                name "joint2A"
              }
            ]
            endPoint Solid {
              translation 0.13 0 0.5
              rotation 1 0 0 0
              children [
                DEF link2AShape Transform {
                  translation 0.07 0 0.2
                  children [
                    Shape {
                      appearance PaintedWood {
                        colorOverride 0.413001 1 0.33489
                      }
                      geometry Box {
                        size 0.1 0.1 0.6
                      }
                    }
                  ]
                }
              ]
              name "link2A"
              boundingObject USE link2AShape
              physics Physics {
                density -1
                mass 0.5
              }
            }
          }
          HingeJoint {
            jointParameters HingeJointParameters {
              anchor 0 0 -0.3
              dampingConstant 0.1
            }
            device [
              PositionSensor {
                name "joint2B"
              }
            ]
            endPoint Solid {
              translation 0.13 0 -0.3
              rotation 1 0 0 0
              children [
                DEF link2BShape Transform {
                  translation 0.07 0 -0.15
                  children [
                    Shape {
                      appearance PaintedWood {
                        colorOverride 0.413001 1 0.33489
                      }
                      geometry Box {
                        size 0.1 0.1 0.4
                      }
                    }
                  ]
                }
              ]
              name "link2B"
              boundingObject USE link2BShape
              physics Physics {
                density -1
                mass 0.5
              }
            }
          }
        ]
        name "link1"
        boundingObject USE link1Shape
        physics Physics {
          density -1
          mass 0.5
        }
      }
    }
  ]
  name "pendulum_1_2"
  boundingObject USE baseObject
  physics Physics {
    density -1
    mass 20
  }
  controller "pendulum_1_2"
}