8be13016bc
Upstream: https://github.com/pollen-robotics/bam Upstream-Commit: 57d13ead53206a6bf0db3d66f86506ae8c2ce01a Upstream-Branch: mjlab_frictionloss
67 lines
1.8 KiB
ReStructuredText
67 lines
1.8 KiB
ReStructuredText
Theoretical Framework
|
|
=====================
|
|
|
|
Pendulum test bench dynamics
|
|
----------------------------
|
|
|
|
For a single-axis pendulum test bench:
|
|
|
|
.. math::
|
|
|
|
\tau_m + \tau_e(\theta) + \tau_f = J\ddot{\theta}
|
|
|
|
where:
|
|
|
|
- :math:`\theta` is the joint position
|
|
- :math:`\tau_m` is the actuator torque
|
|
- :math:`\tau_e(\theta) = mgl\sin(\theta)` is the gravity torque
|
|
- :math:`\tau_f` is the friction torque
|
|
- :math:`J` is the apparent inertia (load + actuator side)
|
|
|
|
Friction as a torque budget
|
|
---------------------------
|
|
|
|
The Coulomb-Viscous form is typically written as a force opposed to velocity.
|
|
In BAM, friction is handled through a maximum available resistive torque
|
|
:math:`\tau_{fm}`.
|
|
|
|
For a discrete simulator with time step :math:`\Delta t`, stopping at next step
|
|
requires:
|
|
|
|
.. math::
|
|
|
|
\dot{\theta}_{k+1} = 0
|
|
|
|
which yields the stopping torque:
|
|
|
|
.. math::
|
|
|
|
\tau_{f,stop} = -\left(\frac{J}{\Delta t}\dot{\theta} + \tau_m + \tau_e\right)
|
|
|
|
The applied friction is then this stopping torque clipped to the available friction budget:
|
|
|
|
.. math::
|
|
|
|
\tau_f = \mathrm{clip}(\tau_{f,stop}, -\tau_{fm}, \tau_{fm})
|
|
|
|
Each model :math:`\mathcal{M}_i` defines :math:`\tau_{fm}` with a different level
|
|
of expressiveness.
|
|
|
|
Servo model + friction model coupling
|
|
-------------------------------------
|
|
|
|
In BAM, simulation combines:
|
|
|
|
- a servo model :math:`\mathcal{S}` that maps state and target to motor torque
|
|
- a friction model :math:`\mathcal{M}` that computes the friction budget
|
|
|
|
At each step:
|
|
|
|
1. compute :math:`\tau_m` from control law and motor model
|
|
2. compute :math:`\tau_{fm}` from chosen friction model
|
|
3. compute :math:`\tau_f` by clipping :math:`\tau_{f,stop}`
|
|
4. integrate acceleration from bench dynamics
|
|
|
|
This makes the framework both identifiable from logs and directly usable in
|
|
simulation.
|