Detailed Description

template<typename T>
class drake::multibody::benchmarks::Acrobot< T >

The Acrobot - a canonical underactuated system as described in Chapter 3 of Underactuated Robotics.

This system essentially is a double pendulum consisting of two links. Link 1 is connected to the world by a "shoulder" revolute joint parameterized by angle theta1 and Link 2 is connected to Link 1 by an "elbow" revolute joint parameterized by angle theta2.

Template Parameters

T	The scalar type, which must be one of the default nonsymbolic scalars.

#include <drake/multibody/benchmarks/acrobot/acrobot.h>

Public Member Functions
	Acrobot (const Vector3< T > &normal, const Vector3< T > &up, double m1=1.0, double m2=1.0, double l1=1.0, double l2=1.0, double lc1=0.5, double lc2=0.5, double Ic1=.083, double Ic2=.33, double b1=0.1, double b2=0.1, double g=9.81)
	Creates an acrobot model in a plane passing through the world's origin and normal to `normal`. More...

Matrix2< T >	CalcMassMatrix (const T &theta2) const
	Computes the mass matrix `H(q)` for the double pendulum system. More...

Vector2< T >	CalcCoriolisVector (const T &theta1, const T &theta2, const T &theta1dot, const T &theta2dot) const
	Computes the bias term `C(q, v) * v` containing Coriolis and gyroscopic effects as a function of the state of the pendulum. More...

Vector2< T >	CalcGravityVector (const T &theta1, const T &theta2) const
	Computes the effective joint-space torques induced by gravity `tau_g(q)` containing the effect of gravity as a function of the configuration of the pendulum. More...

math::RigidTransform< T >	CalcLink1PoseInWorldFrame (const T &theta1) const
	Computes the pose of the center of mass of link 1 measured and expressed in the world frame. More...

math::RigidTransform< T >	CalcLink2PoseInWorldFrame (const T &theta1, const T &theta2) const
	Computes the pose of the center of mass of link 2 measured and expressed in the world frame. More...

math::RigidTransform< T >	CalcElbowOutboardFramePoseInWorldFrame (const T &theta1, const T &theta2) const
	Computes the pose of the elbow outboard frame `Eo` in the world frame W. More...

Vector6< T >	CalcLink1SpatialVelocityInWorldFrame (const T &theta1, const T &theta1dot) const
	Computes the spatial velocity of the center of mass of link 1 expressed in the world frame. More...

Vector6< T >	CalcLink2SpatialVelocityInWorldFrame (const T &theta1, const T &theta2, const T &theta1dot, const T &theta2dot) const
	Computes the spatial velocity of the center of mass of link 2 expressed in the world frame. More...

Vector6< T >	CalcLink1SpatialAccelerationInWorldFrame (const T &theta1, const T &theta1dot, const T &theta1dotdot) const
	Computes the spatial acceleration of the center of mass of link 1 expressed in the world frame. More...

Vector6< T >	CalcLink2SpatialAccelerationInWorldFrame (const T &theta1, const T &theta2, const T &theta1dot, const T &theta2dot, const T &theta1dotdot, const T &theta2dotdot) const
	Computes the spatial acceleration of the center of mass of link 2 expressed in the world frame. More...

T	CalcPotentialEnergy (const T &theta1, const T &theta2) const
	Computes the total potential energy due to gravity of the acrobot system for the state given by angles `theta1` and `theta2`. More...

Constructor & Destructor Documentation

◆ Acrobot()

Acrobot	(	const Vector3< T > &	normal,
		const Vector3< T > &	up,
		double	m1 = `1.0`,
		double	m2 = `1.0`,
		double	l1 = `1.0`,
		double	l2 = `1.0`,
		double	lc1 = `0.5`,
		double	lc2 = `0.5`,
		double	Ic1 = `.083`,
		double	Ic2 = `.33`,
		double	b1 = `0.1`,
		double	b2 = `0.1`,
		double	g = `9.81`
	)

Creates an acrobot model in a plane passing through the world's origin and normal to normal.

Vector up defines the upwards direction on this plane. Both normal and up are expressed in the world's frame. Essentially the two dimensional equations of the acrobot are described in a model frame D within a x-y plane with y the vertical direction and gravity pointing downwards. Therefore the axes defining the model frame D are:

  z_W = normal_W.normalized()
  y_W = (up - up.dot(z_W) * z_W).normalized()
  x_W = y_W.cross(z_W)

The remaining arguments define the properties of the double pendulum system:

m1: mass of the first link.
m2: mass of the second link.
l1: length of the first link.
l2: length of the second link.
lc1: length from the shoulder to the center of mass of the first link.
lc2: length from the elbow to the center of mass of the second link.
Ic1: moment of inertia about the center of mass for the first link.
Ic2: moment of inertia about the center of mass for the second link.
b1: damping coefficient of the shoulder joint.
b2: damping coefficient of the elbow joint.
g: acceleration of gavity.

Member Function Documentation

◆ CalcCoriolisVector()

Vector2<T> CalcCoriolisVector	(	const T &	theta1,
		const T &	theta2,
		const T &	theta1dot,
		const T &	theta2dot
	)		const

Computes the bias term C(q, v) * v containing Coriolis and gyroscopic effects as a function of the state of the pendulum.

◆ CalcElbowOutboardFramePoseInWorldFrame()

math::RigidTransform<T> CalcElbowOutboardFramePoseInWorldFrame	(	const T &	theta1,
		const T &	theta2
	)		const

Computes the pose of the elbow outboard frame Eo in the world frame W.

Parameters

theta1	The shoulder angle in radians.
theta2	The elbow angle in radians.

Returns: X_WEo the pose of the elbow frame Eo in the world frame W.

◆ CalcGravityVector()

Vector2<T> CalcGravityVector	(	const T &	theta1,
		const T &	theta2
	)		const

Computes the effective joint-space torques induced by gravity tau_g(q) containing the effect of gravity as a function of the configuration of the pendulum.

Unlike http://underactuated.mit.edu.ezproxy.canberra.edu.au/underactuated.html?chapter=3, cited in this class's documentation, we define tau_g(q) to be on the right hand side of the equations of motion, that is, Mv̇ + C(q, v)v = tau_g(q).

◆ CalcLink1PoseInWorldFrame()

math::RigidTransform<T> CalcLink1PoseInWorldFrame ( const T & theta1 ) const

Computes the pose of the center of mass of link 1 measured and expressed in the world frame.

Parameters

theta1	The shoulder angle in radians.
theta2	The elbow angle in radians.

Returns: X_WL1 the pose of link 1 measured and expressed in the world frame.

◆ CalcLink1SpatialAccelerationInWorldFrame()

Vector6<T> CalcLink1SpatialAccelerationInWorldFrame	(	const T &	theta1,
		const T &	theta1dot,
		const T &	theta1dotdot
	)		const

Computes the spatial acceleration of the center of mass of link 1 expressed in the world frame.

Parameters

theta1	The shoulder angle in radians.
theta1dot	The shoulder angular velocity in radians per second.
theta1dotdot	The elbow angular acceleration in radians per second squared.

Return values

A_WL1_W the spatial acceleration of the center of mass of link 1 with respect to the world and expressed in the world frame.

◆ CalcLink1SpatialVelocityInWorldFrame()

Vector6<T> CalcLink1SpatialVelocityInWorldFrame	(	const T &	theta1,
		const T &	theta1dot
	)		const

Computes the spatial velocity of the center of mass of link 1 expressed in the world frame.

Parameters

theta1	The shoulder angle in radians.
theta1dot	The shoulder angular velocity in radians per second.

Returns: V_WL1_W the spatial velocity of the center of mass of link 1 with respect to the world and expressed in the world frame.

◆ CalcLink2PoseInWorldFrame()

math::RigidTransform<T> CalcLink2PoseInWorldFrame	(	const T &	theta1,
		const T &	theta2
	)		const

Computes the pose of the center of mass of link 2 measured and expressed in the world frame.

Parameters

theta1	The shoulder angle in radians.
theta2	The elbow angle in radians.

Returns: X_WL2 the pose of link 2 measured and expressed in the world frame.

◆ CalcLink2SpatialAccelerationInWorldFrame()

Vector6<T> CalcLink2SpatialAccelerationInWorldFrame	(	const T &	theta1,
		const T &	theta2,
		const T &	theta1dot,
		const T &	theta2dot,
		const T &	theta1dotdot,
		const T &	theta2dotdot
	)		const

Computes the spatial acceleration of the center of mass of link 2 expressed in the world frame.

Parameters

theta1	The shoulder angle in radians.
theta2	The elbow angle in radians.
theta1dot	The shoulder angular velocity in radians per second.
theta2dot	The elbow angular velocity in radians per second.
theta1dotdot	The shoulder angular acceleration in radians per second squared.
theta2dotdot	The elbow angular acceleration in radians per second squared.

Return values

A_WL2_W the spatial acceleration of the center of mass of link 2 with respect to the world and expressed in the world frame.

◆ CalcLink2SpatialVelocityInWorldFrame()

Vector6<T> CalcLink2SpatialVelocityInWorldFrame	(	const T &	theta1,
		const T &	theta2,
		const T &	theta1dot,
		const T &	theta2dot
	)		const

Computes the spatial velocity of the center of mass of link 2 expressed in the world frame.

Parameters

theta1	The shoulder angle in radians.
theta2	The elbow angle in radians.
theta1dot	The shoulder angular velocity in radians per second.
theta2dot	The elbow angular velocity in radians per second.

Returns: V_WL2_W the spatial velocity of the center of mass of link 2 with respect to the world and expressed in the world frame.

◆ CalcMassMatrix()

Matrix2<T> CalcMassMatrix ( const T & theta2 ) const

Computes the mass matrix H(q) for the double pendulum system.

It turns out that for this system the mass matrix is independent of the shoulder angle theta1.

◆ CalcPotentialEnergy()

T CalcPotentialEnergy	(	const T &	theta1,
		const T &	theta2
	)		const

Computes the total potential energy due to gravity of the acrobot system for the state given by angles theta1 and theta2.

The zero potential energy is defined for y = 0.

The documentation for this class was generated from the following file:

drake/multibody/benchmarks/acrobot/acrobot.h

Detailed Description

template<typename T> class drake::multibody::benchmarks::Acrobot< T >

Public Member Functions

Constructor & Destructor Documentation

◆ Acrobot()

Member Function Documentation

◆ CalcCoriolisVector()

◆ CalcElbowOutboardFramePoseInWorldFrame()

◆ CalcGravityVector()

◆ CalcLink1PoseInWorldFrame()

◆ CalcLink1SpatialAccelerationInWorldFrame()

◆ CalcLink1SpatialVelocityInWorldFrame()

◆ CalcLink2PoseInWorldFrame()

◆ CalcLink2SpatialAccelerationInWorldFrame()

◆ CalcLink2SpatialVelocityInWorldFrame()

◆ CalcMassMatrix()

◆ CalcPotentialEnergy()

template<typename T>
class drake::multibody::benchmarks::Acrobot< T >