Concepts

This section explains the key concepts and conventions used in the DexRobot Kinematics library.

Coordinate Systems and Frames

The library uses two primary reference frames:

  1. Hand Frame: Local coordinate system centered at the hand base.

    • Origin at the hand base/wrist

    • X-axis pointing toward palm

    • Y-axis pointing away from thumb for left hand, toward thumb for right hand

    • Z-axis pointing toward fingertips

    Hand Frame

  2. World Frame: Global coordinate system. When using a hand in isolation:

    • The hand frame aligns with the world frame unless a base_pose is specified

    • With a base_pose, all computations transform between these frames

┌─────────┐  base_pose   ┌──────────┐
│ World   │ ───────────> │ Hand     │
│ Frame   │ <─────────── │ Frame    │
└─────────┘              └──────────┘

The hand can be placed anywhere in the world frame by specifying a base_pose:

import numpy as np
import pinocchio as pin
from dexrobot_kinematics.utils.types import Position, Pose

# Create rotation matrix (90 degrees around Y axis)
R = pin.utils.rpyToMatrix(0, np.pi/2, 0)

# Create translation vector (1 meter in X direction)
t = np.array([1.0, 0.0, 0.0])

# Define base pose
base_pose = Pose(position=Position.from_array(t), orientation=R)

# Use in forward kinematics
poses = hand.forward_kinematics(joint_angles, base_pose=base_pose, frame="world")

The same base_pose can be used in inverse kinematics to transform target positions from the world frame to the hand frame.

End Effector Types

The library supports two types of end effectors:

  1. Fingerpads: The contact surfaces on each finger, used for grasping objects (green markers in the image below).

  2. Fingertips: The very ends of each finger (red markers in the image below).

End Effectors

You can specify which end effector type to use in both forward and inverse kinematics:

# Get fingerpad poses
fingerpad_poses = hand.forward_kinematics(joint_angles, end_effector="fingerpad")

# Get fingertip poses
fingertip_poses = hand.forward_kinematics(joint_angles, end_effector="fingertip")

Inverse Kinematics Principles

The IK solver uses a damped least-squares method (Levenberg-Marquardt) to iteratively solve for joint angles:

  1. Objective: Find joint angles that place end effector(s) at target position(s)

  2. Method: - Start with initial guess for joint angles - Compute current end effector position - Calculate error vector (difference between current and target position) - Compute Jacobian matrix (relates joint velocities to end effector velocities) - Update joint angles using damped least-squares formula - Repeat until convergence or maximum iterations

The solver handles constraints including: - Joint limits - Joint coupling between PIP and DIP joints, consistent with the hand’s mechanical design