Regularization#
Default L2 regularization#
Specified tasks might lead to an under-constrained problem, where an infinite number of solutions exist. In that case, the solver will try to minimize the norm of the configuration variation, which will lead to a solution that is as close as possible to the current configuration.
This is because a very low weighted cost is always present in front of the norm of the configuration variation \(\lVert \Delta q \rVert^2\).
6-axis regularization (default)
In this example, the robot is following a target 3D position. The problem is under-constrained, so the default regularization will apply.
Custom L2 regularization#
You can add your own regularization by adding a RegularizationTask:
# Adding a custom regularization task
regularization_task = solver.add_regularization_task(1e-4)
The greater is the provided regularization value, the more expansive it will be for the solver to move any joint.
6-axis regularization (custom strong L2)
If you pass --strong_l2 to the following example, a strong \(10^{2}\) L2 regularization will be used.
This will create a strong cost against the motion and cause the robot to lag behind its target.
Posture regularization#
Another way to regularize the problem is using a posture regularization. This can be achieved with e.g
a JointsTask:
# Adding a posture regularization task
joints_task = solver.add_joints_task()
joints_task.set_joints({
joint: 0.0
for joint in robot.joint_names()
})
joints_task.configure("posture", "soft", 1e-6)
Here, the joints task is setting a target to zero for all joints with a small soft weight of 1e-5. When no tasks are specified, the joints will go back to those positions.
6-axis regularization (posture)
If you pass --posture to the following example, a joint task with \(10^{-6}\) weight will be used.
Note that the pose of the robot is slightly different from the default regularization
Kinetic energy regularization#
Another possible regularization is to minimize the kinetic energy in the system. This can be done by using
a KineticEnergyRegularizationTask:
# Adding a kinetic energy regularization task
kinetic_energy_task = solver.add_kinetic_energy_regularization_task(1e-6)
This will minimise \(\frac{1}{2} \dot{q}^T M \dot{q}\), where \(M\) is the inertia matrix of the robot.
Note
When using this regularization, and to ensure that the tas has the unit of a kinetic energy,
you have to specify the \(\Delta t\) between two successive calls to the solver.
This can be done by setting solver.dt:
# Setting solver.dt is required to use kinetic energy regularization
solver.dt = 0.01
6-axis regularization (kinetic energy)
If you pass --kinetic to the following example, a kinetic energy regularization task with \(10^{-6}\)
weight will be used. Since the energy is weighted by the mass matrix, the end of the arm will
be more used than the base.