Decomposition and Cross-Product-Based Method for Computing the Dynamic Equation of Robots

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Introduction
When looking at robotics research, dynamic modelling is basically a form of artwork. The complexity of a dynamic model concerns the degrees of freedom (DOFs) of a machine or a robot and the dynamic model of a low DOFs robot can be easily developed, but dynamic models for a high DOFs robot are difficult to derive by hand. The drawbacks to creating the models by hand are: difficulty of maintenance, complexity of proofreading the dynamic model and inflexibility. The biped robot is the target of a complex dynamic system study [1] that also reveals a demand for a computer-aided systematic derivation tools for robotics.
Using an auto-deriving tool to obtain a dynamic model of a robot is very economical. Tošic [2] proposed the benefits of a symbolic simulation of an engineering system, while Cetinkunt [3] proposed a symbolic modelling of the dynamics of robotic manipulators on the numerical tool REDUCE. Since the coding of REDUCE is based on C and Fortran, modifying the system model by simply changing the symbol is rather difficult. Vibet demonstrated the symbolic tool FORM [4], which automatically generates the dynamic equations of the manipulator. The MACSYMA tool was then introduced to mechanical engineering education [5], providing the steps for determining a system dynamics model, but it does not support online simulations that integrate both system models and control rules.
Numerous contributions in both algorithms and their computational efficiency have been made in the field of robot dynamics [6][7][8][9]. The need for an online computing [6] method is motivated by simulation studies of the dynamics of a robotic system with large DOFs. While computation efficiency continues to be crucial for the simulation and control of increasingly complex mechanisms operating at higher speeds, other aspects of the dynamics problem are also vital. Algorithms should be formulated under a simple framework to enable ease of development, modification, extension and understanding.
This paper aims to demonstrate a clear relationship between Lagrange equations and Newton-Euler equations regarding the computational methods for robot dynamics, from which we derive a systematic method using either symbolic or on-line numerical computations. First, this study reviews the Lagrange formulation, as well as its symbolic generation method, which is suitable for studying a low-DOF robotic system. Second, based on the decomposition approach and cross-product operation, a computing framework can be easily developed to calculate the inertia matrix, the Coriolis and centrifugal matrix, and the gravity force vector of robot dynamics equations. The advantages of this computing method are that: it can be used for both symbolic and the on-line numeric computation purposes, and it also can be applied to a closed-chain robot system. We present several robotic system examples to demonstrate the versatility of the proposed method.

Decomposition method
Assume that a robot system has a generalized coordinate with n DOFs and let The equation of robot motion [9] can be written as is a skew-symmetric matrix caused by the null energy property.

Lagrange formulation approach
The total kinetic energy K of a robot consists of the rotational kinetic energy i K  and the translational kinetic energy vi K of each link. The total potential energy P consists of the potential energy i P of each link; and therefore, and    n i i P P 1 . ( The kinetic energies The potential energy of link i can be obtained by where g is the acceleration of gravity and where q p J ci vi    is the linear Jacobian matrix of link i, and the translational energy of link i is via From the Lagrange formulation, we have (11)

Newton-Euler formulation approach
The decomposition of the robot's equations of motion can also be conducted using the Newton-Euler approach. The angular acceleration of link i is expressed as and hence, which is the same as Therefore, i D  and i C  can be obtained respectively using: The acceleration of CoM of link i is which is also the same as (20)

Extended coordinate system
The Cartesian coordinates of the base point,  where n is the number of joints. For simplicity, it is assumed that the relative motion of link i with respect to link (i-1) is either the rotational (revolute joint) or the translational (prismatic joint) motion with one of the basic axes in the local body frame from the set {  x-axis,  y-axis, and  z-axis } and is denoted by Define three basic rotational matrices: The procedures for computing the dynamic equation of a serial-link robot are thus as follows: Step 1: Computing the rotational matrix of link i, i R ,

Tz
where ) (:,k R , 3 , 2 , 1  k represents the k-th column of the matrix R .
Step 2: Computing the angular Jacobian matrix of link i, The angular Jacobian matrix of link i, where ) (:, j J i  represents the j-th column of matrix i J  and the total rotational kinetic energy  K is obtained from Step 3: Differentiation of the angular Jacobian matrices The angular acceleration of link i is obtained as The total translational kinetic energy v K and the total potential energy P are obtained as 1 1 Step 6: Putting it all together The D , C , and G matrices of the robot's equations of For a link with two joints, a virtual link with zero mass and/or zero inertia can be inserted for easy programming. The proposed method and the Lagrange method are shown to obtain the same equations of motion as follows   Table 1. System parameters of the 3-DOF spherical robot system.    Table 2. System parameters of a 3-DOF robotic pendulum system. Table 3. System parameters of a biped robot.

Example 3: a 10-DOF biped robot
This example considers a 10-DOF fully actuated 3D biped with feet, as shown in Figure 3. The 3D biped robot consists of 7 links: a torso and two legs and two feet. Each leg has a knee and an equal length for the shin and thigh. The 10 actuators are installed as follows: two at the hip, one at the knee and two at the ankle.
is the centre of mass of the i-th link, is the moment of the i-th link. For authentication, the 10-DOF biped robot online simulation work has been implemented on MATLAB. After symbolic calculation, we transfer the dynamic equations into C-Mex files to improve the efficiency of the biped robot walking simulation.

Example 4: a simple 2-DOF closed-chain robot
We can also apply the proposed method to calculate the equations of motion for simple closed-chain robotic systems. Figure 4 shows

Conclusion
This paper has developed a decomposition and cross-product-based method to compute robot dynamics, offering several examples that demonstrate the versatility of the proposed method. The focus of this paper is first to establish a clear relationship between Lagrange equations and Newton-Euler equations, and second to develop an extremely intuitive and systematic method to compute the inertia matrix, the Coriolis and centrifugal matrix, and the gravity force vector of robot dynamics equations.