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fifth joint of the robot device has a lightweight mechanical structure and works in the moment mode with
almost zero resistance, the kinetic model is not established for this joint, and this operation can reduce the
complexity of the robot model and improve the model accuracy. Therefore, the dynamics model of this robot
is a 4-degree-of-freedom model.

2.2.1 Kinetic model construction
The Newton-Euler method is used to build a dynamics model for this robot. Compared with the Lagrangian
modellingmethod, thisapproachcalculatesthroughtheinter-jointforceandmotionrelationshiplayerbylayer,
the physical meaning is clear, and it does not involve derivation operation, so that the results can be obtained
quickly and the calculation efficiency is high [34] . The standard form of its expression is given as

= ( ) ¥ + ( , ¤) + ( ) + ( , ¤, ¥) (3)
where , ¤, ¥ represent the position, velocity and acceleration vectors of the joint, respectively, denotes the

joint torque vector, ( ) is the mass matrix, ( , ¤) is the centrifugal and Koch force vectors, ( ) denotes

the gravitational moment, and ( , ¤, ¥) denotes the other compensating moment parameters.

2.2.2 Linearisation of the model
The kinetic parameters of the robot are represented in the kinetic model in a nonlinear combination, which
makesitdifficulttoidentifythekineticparameters. Throughtheparallelaxistheorem,thecoordinatesystemof
theinertialparameterinthenonlineartermiscoordinatetransformedtocompletethelinearisationprocess [35] .
The force and moment expressions in the kinetic model are given in
+1
= +1 +1 + ¤ × + × × + ¤
+1 +1 (4)
= +1 + +1 × × ¤ + × + ¤
+1 +1 +1 +
where , are the force and moment on the articulated linkage, +1 is the coordinate transformation matrix,

¤ , , ¤ represents the linear acceleration, angular velocity, and angular acceleration of the articulated linkage

of the rehabilitation robot, respectively, +1 is the vector from the origin of the coordinate system of the ℎ
articulated linkage (i.e., the coordinate system) to the origin of the coordinate system of the +1 ℎ articulated

linkage, isthemassofthearticulatedlinkage, and representsthecentreofmassofthearticulatedlinkage.
denotes the inertia tensor matrix. The identification equation for the kinetic parameters can be expressed as:

= ( , ¤, ¥) (5)

where ( , ¤, ¥) ∈ × denotes the observation matrix; and denote the number of kinetic parameters

in the kinetic parameter set and the number of robot rods, respectively. ∈ ×1 is the set of dynamics
parameters. Since some columns of ( , ¤, ¥) are always zero and some columns have a linear relationship,

no matter what value of , ¤, ¥ is taken, cannot make the columns full rank, and thus cannot be solved

uniquely by the least-squares method. In this section, we use the decomposition method. The matrix
is decomposed to full rank, and the result of the decomposition is used to restructure the inertia parameters.
The following formula can be obtained:
˜ (6)
=
˜
where is the matrix of column full-rank coefficients obtained from the full-rank decomposition of .
is the minimum inertia parameter after reorganisation.
2.2.3 Fourier excitation trajectory
In the realm of robot dynamics parameter identification, the judicious linearisation of the model ensures the
unique convergence of identification results toward the target values. Discrepancies between target and true
values primarily stem from kinematic parameter deviations and measurement noise. Rational design of iden-
tification excitation trajectories serves to mitigate the impact of measurement noise on results and enhance