Package: pin-pink
PinkPythoninversekinematics for articulated robot models, based onPinocchio.Installationpip install pin-pinkUsagePink solves differential inverse kinematics byweighted tasks. A task is defined by aresidualfunction $e(q)$ of the robot configuration $q \in \mathcal{C}$ to be driven to zero. For instance, putting a foot position $p_{foot}(q)$ at a given target $p_{foot}^{\star}$ can be described by the position residual:$$
e(q) = p_{foot}^{\star} - p_{foot}(q)
$$In differential inverse kinematics, we compute a velocity $v \in \mathfrak{c}$ that satisfies the first-order differential equation:$$
J_e(q) v = \dot{e}(q) = -\alpha e(q)
$$where $J_e(q) := \frac{\partial e}{\partial q}$ is thetask Jacobian. We can define multiple tasks, but some of them will come into conflict if they can't be all fully achieved at the same time. Conflicts are resolved by casting all objectives to a common unit, and weighing these normalized objectives relative to each other. We also include configuration and velocity limits, making our overall optimization problem a quadratic program:$$
\begin{align}
\underset{v \in \mathfrak{c}}{\text{minimize}} \ & \sum_{\text{task } e} \Vert J_e(q) v + \alpha e(q) \Vert^2_{W_e} \
\text{subject to} \ & v_{\text{min}}(q) \leq v \leq v_{\text{max}}(q)
\end{align}
$$Pink provides an API to describe the problem as tasks with targets, and automatically build and solve the underlying quadratic program.Task costsHere is the example of a biped robot that controls the position and orientation of its base, left and right contact frames. A fourth "posture" task, giving a preferred angle for each joint, is added for regularization:frompink.tasksimportFrameTask,PostureTasktasks={"base":FrameTask("base",position_cost=1.0,# [cost] / [m]orientation_cost=1.0,# [cost] / [rad]),"left_contact":FrameTask("left_contact",position_cost=[0.1,0.0,0.1],# [cost] / [m]orientation_cost=0.0,# [cost] / [rad]),"right_contact":FrameTask("right_contact",position_cost=[0.1,0.0,0.1],# [cost] / [m]orientation_cost=0.0,# [cost] / [rad]),"posture":PostureTask(cost=1e-3,# [cost] / [rad]),}Orientation (similarly position) costs can be scalars or 3D vectors. They specify how much each radian of angular error "costs" in the overall normalized objective. When using 3D vectors, components are weighted anisotropically along each axis of the body frame.Task targetsAside from their costs, most tasks take a second set of parameters calledtarget. For example, a body task aims for a target transform, while a posture task aims for a target configuration vector. Targets are set by theset_targetfunction:tasks["posture"].set_target([1.0,0.0,0.0,0.0]+# floating base quaternion[0.0,0.0,0.0]+# floating base position[0.0,0.2,0.0,0.0,-0.2,0.0]# joint angles)Body tasks can be initialized, for example, from the robot's neutral configuration:importpinkfromrobot_descriptions.loaders.pinocchioimportload_robot_descriptionrobot=load_robot_description("upkie_description")configuration=pink.Configuration(robot.model,robot.data,robot.q0)forbody,taskintasks.items():iftype(task)isFrameTask:task.set_target(configuration.get_transform_frame_to_world(body))A task can be added to the inverse kinematics once both its cost and target (if applicable) are defined.Differential inverse kinematicsPink solves differential inverse kinematics, meaning it outputs a velocity that steers the robot towards achieving all tasks at best. If we keep integrating that velocity, and task targets don't change over time, we will converge to a stationary configuration:dt=6e-3# [s]fortinnp.arange(0.0,42.0,dt):velocity=solve_ik(configuration,tasks.values(),dt,solver="quadprog")configuration.integrate_inplace(velocity,dt)time.sleep(dt)If task targets are continuously updated, there will be no stationary solution to converge to, but the model will keep on tracking each target at best. Note thatsolve_ikwill take care of both configuration and velocity limits read from the robot model.ExamplesBasic examples to get started:Double pendulumLoading a custom URDFVisualization in MeshCatVisualization in yourdfpyPink works with all kinds of robot morphologies:Arms:Kinova Gen2,UR3Humanoids:JVRC-1,SigmaBanMobile base:Omnidirectional robot,Stretch R1Wheeled biped:UpkieCheck out theexamplesfolder for more.How can I help?Install the library and use it! Report bugs in theissue tracker. If you are a developer with some robotics experience looking to hack on open source, check out thecontribution guidelines.See alsoSoftware:Jink.jl: Julia package for differential multi-task inverse kinematics.pymanoid: precursor to Pink based on OpenRAVE.Technical notes:Inverse kinematics: a general introduction to differential inverse kinematics.Jacobian of a kinematic task and derivatives on manifolds.