Motion Generators ================= Robotic tasks are typically defined in task-space in terms of desired end-effector trajectory, while control actions are executed in the joint-space. This naturally leads to *joint-space* and *task-space* (operational-space) control methods. However, successful execution of interaction tasks using motion control often requires an accurate model of both the robot manipulator as well as its environment. While a sufficiently precise manipulator's model might be known, detailed description of environment is hard to obtain :cite:p:`siciliano2009force`. Planning errors caused by this mismatch can be overcome by introducing a *compliant* behavior during interaction. While compliance is achievable passively through robot's structure (such as elastic actuators, soft robot arms), we are more interested in controller designs that focus on active interaction control. These are broadly categorized into: 1. **impedance control:** indirect control method where motion deviations caused during interaction relates to contact force as a mass-spring-damper system with adjustable parameters (stiffness and damping). A specialized case of this is *stiffness* control where only the static relationship between position error and contact force is considered. 2. **hybrid force/motion control:** active control method which controls motion and force along unconstrained and constrained task directions respectively. Among the various schemes for hybrid motion control, the provided implementation is based on inverse dynamics control in the operational space :cite:p:`khatib1987osc`. .. note:: To provide an even broader set of motion generators, we welcome contributions from the community. If you are interested, please open an issue to start a discussion! Joint-space controllers ----------------------- Torque control ~~~~~~~~~~~~~~ Action dimensions: ``"n"`` (number of joints) In torque control mode, the input actions are directly set as feed-forward joint torque commands, i.e. at every time-step, .. math:: \tau = \tau_{des} Thus, this control mode is achievable by setting the command type for the actuator group, via the :class:`ActuatorControlCfg` class, to ``"t_abs"``. Velocity control ~~~~~~~~~~~~~~~~ Action dimensions: ``"n"`` (number of joints) In velocity control mode, a proportional control law is required to reduce the error between the current and desired joint velocities. Based on input actions, the joint torques commands are computed as: .. math:: \tau = k_d (\dot{q}_{des} - \dot{q}) where :math:`k_d` are the gains parsed from configuration. This control mode is achievable by setting the command type for the actuator group, via the :class:`ActuatorControlCfg` class, to ``"v_abs"`` or ``"v_rel"``. .. attention:: While performing velocity control, in many cases, gravity compensation is required to ensure better tracking of the command. In this case, we suggest disabling gravity for the links in the articulation in simulation. Position control with fixed impedance ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Action dimensions: ``"n"`` (number of joints) In position control mode, a proportional-damping (PD) control law is employed to track the desired joint positions and ensuring the articulation remains still at the desired location (i.e., desired joint velocities are zero). Based on the input actions, the joint torque commands are computed as: .. math:: \tau = k_p (q_{des} - q) - k_d \dot{q} where :math:`k_p` and :math:`k_d` are the gains parsed from configuration. In its simplest above form, the control mode is achievable by setting the command type for the actuator group, via the :class:`ActuatorControlCfg` class, to ``"p_abs"`` or ``"p_rel"``. However, a more complete formulation which considers the dynamics of the articulation would be: .. math:: \tau = M \left( k_p (q_{des} - q) - k_d \dot{q} \right) + g where :math:`M` is the joint-space inertia matrix of size :math:`n \times n`, and :math:`g` is the joint-space gravity vector. This implementation is available through the :class:`JointImpedanceController` class by setting the impedance mode to ``"fixed"``. The gains :math:`k_p` are parsed from the input configuration and :math:`k_d` are computed while considering the system as a decoupled point-mass oscillator, i.e., .. math:: k_d = 2 \sqrt{k_p} \times D where :math:`D` is the damping ratio of the system. Critical damping is achieved for :math:`D = 1`, overcritical damping for :math:`D > 1` and undercritical damping for :math:`D < 1`. Additionally, it is possible to disable the inertial or gravity compensation in the controller by setting the flags :attr:`inertial_compensation` and :attr:`gravity_compensation` in the configuration to :obj:`False`, respectively. Position control with variable stiffness ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Action dimensions: ``"2n"`` (number of joints) In stiffness control, the same formulation as above is employed, however, the gains :math:`k_p` are part of the input commands. This implementation is available through the :class:`JointImpedanceController` class by setting the impedance mode to ``"variable_kp"``. Position control with variable impedance ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Action dimensions: ``"3n"`` (number of joints) In impedance control, the same formulation as above is employed, however, both :math:`k_p` and :math:`k_d` are part of the input commands. This implementation is available through the :class:`JointImpedanceController` class by setting the impedance mode to ``"variable"``. Task-space controllers ---------------------- Differential inverse kinematics (IK) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Action dimensions: ``"3"`` (relative/absolute position), ``"6"`` (relative pose), or ``"7"`` (absolute pose) Inverse kinematics converts the task-space tracking error to joint-space error. In its most typical implementation, the pose error in the task-sace, :math:`\Delta \chi_e = (\Delta p_e, \Delta \phi_e)`, is computed as the cartesian distance between the desired and current task-space positions, and the shortest distance in :math:`\mathbb{SO}(3)` between the desired and current task-space orientations. Using the geometric Jacobian :math:`J_{eO} \in \mathbb{R}^{6 \times n}`, that relates task-space velocity to joint-space velocities, we design the control law to obtain the desired joint positions as: .. math:: q_{des} = q + \eta J_{eO}^{-} \Delta \chi_e where :math:`\eta` is a scaling parameter and :math:`J_{eO}^{-}` is the pseudo-inverse of the Jacobian. It is possible to compute the pseudo-inverse of the Jacobian using different formulations: * Moore-Penrose pseduo-inverse: :math:`A^{-} = A^T(AA^T)^{-1}`. * Levenberg-Marquardt pseduo-inverse (damped least-squares): :math:`A^{-} = A^T (AA^T + \lambda \mathbb{I})^{-1}`. * Tanspose pseudo-inverse: :math:`A^{-} = A^T`. * Adaptive singular-vale decomposition (SVD) pseduo-inverse from :cite:t:`buss2004ik`. These implementations are available through the :class:`DifferentialInverseKinematics` class. Impedance controller ~~~~~~~~~~~~~~~~~~~~ It uses task-space pose error and Jacobian to compute join torques through mass-spring-damper system with a) fixed stiffness, b) variable stiffness (stiffness control), and c) variable stiffness and damping (impedance control). Operational-space controller ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Similar to task-space impedance control but uses the Equation of Motion (EoM) for computing the task-space force Closed-loop proportional force controller ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It uses a proportional term to track the desired wrench command with respect to current wrench at the end-effector. Hybrid force-motion controller ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It combines closed-loop force control and operational-space motion control to compute the desired wrench at the end-effector. It uses selection matrices that define the unconstrainted and constrained task directions. Reactive planners ----------------- Typical task-space controllers do not account for motion constraints such as joint limits, self-collision and environment collision. Instead they rely on high-level planners (such as RRT) to handle these non-Euclidean constraints and give joint/task-space way-points to the controller. However, these methods are often conservative and have undesirable deceleration when close to an object. More recently, different approaches combine the constraints directly into an optimization problem, thereby providing a holistic solution for motion generation and control. We currently support the following planners: - **RMPFlow (lula):** An acceleration-based policy that composes various Reimannian Motion Policies (RMPs) to solve a hierarchy of tasks :cite:p:`cheng2021rmpflow`. It is capable of performing dynamic collision avoidance while navigating the end-effector to a target. - **MPC (OCS2):** A receding horizon control policy based on sequential linear-quadratic (SLQ) programming. It formulates various constraints into a single optimization problem via soft-penalties and uses automatic differentiation to compute derivatives of the system dynamics, constraints and costs. Currently, we support the MPC formulation for end-effector trajectory tracking in fixed-arm and mobile manipulators. The formulation considers a kinematic system model with joint limits and self-collision avoidance :cite:p:`mittal2021articulated`. .. warning:: We wrap around the python bindings for these reactive planners to perform a batched computing of robot actions. However, their current implementations are CPU-based which may cause certain slowdown for learning.