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Robot Model

Kinematics

First define the home configuration as pointing straight up and all joint positions at zero as a homogenous transformation matrix
.
Define the screws for each joint
where
is the joint axis of rotation and
corresponds to the base-frame joint origins in the home configuration.
Use the product of exponentials to compute the forward kinemtics
for joints i = 1 to n
Use the adjoint
Where
is the adjoint transformation of the homogenous transform for frame i

Dynamics

Define the robot dynamics as
where

In Software

Export the CAD to URDF
Get FK and dynamics via pinnochio
import pinocchio as pin
from pinocchio.urdf import buildModel

model = buildModel("ur10e.urdf.xacro")
data = model.createData()

# Forward kinematics
q = pin.neutral(model)
pin.forwardKinematics(model, data, q)

# Dynamics
M = pin.crba(model, data, q)
G = pin.computeGeneralizedGravity(model, data, q)
C = pin.computeCoriolisMatrix(model, data, q, np.zeros(model.nv))

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